College Algebra Exam 1 Review: Step-by-Step Study Guidance

{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1. Evaluating Contextual Functions"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Function evaluation and interpretation in context."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to interpret function notation in a real-world scenario and to evaluate a linear function for a given input."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Function notation: "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" represents the output value when input "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" is used."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Linear function: "},{"type":"inlineMath","attrs":{"latex":"R(d) = 0.85d + 40"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For part (a), identify what "},{"type":"inlineMath","attrs":{"latex":"f(5) = 18"}},{"type":"text","text":" means: "},{"type":"inlineMath","attrs":{"latex":"t = 5"}},{"type":"text","text":" years, "},{"type":"inlineMath","attrs":{"latex":"V = 18"}},{"type":"text","text":" (in thousands of dollars)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write a sentence that connects the input (years since purchase) and the output (van's value in thousands)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For part (b), substitute "},{"type":"inlineMath","attrs":{"latex":"d = 80"}},{"type":"text","text":" into the revenue function: "},{"type":"inlineMath","attrs":{"latex":"R(80) = 0.85 \\times 80 + 40"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate "},{"type":"inlineMath","attrs":{"latex":"0.85 \\times 80"}},{"type":"text","text":" and then add $40$ to that result."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Interpret what the output of "},{"type":"inlineMath","attrs":{"latex":"R(80)"}},{"type":"text","text":" means in the context of the van's rental revenue."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) At 5 years after purchase, the estimated value of the van is $18,000."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) "},{"type":"inlineMath","attrs":{"latex":"R(80) = 0.85 \\times 80 + 40 = 68 + 40 = 108"}},{"type":"text","text":". Driving the van 80 miles results in a daily rental revenue of $108."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Evaluating and Graphing Functions"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Evaluating quadratic functions and distinguishing between linear and non-linear functions."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question checks your ability to substitute values into a quadratic function and to classify the function's type based on its algebraic form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quadratic function: "},{"type":"inlineMath","attrs":{"latex":"h(x) = -x^2 + 5x + 12"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Linear function: "},{"type":"inlineMath","attrs":{"latex":"f(x) = mx + b"}},{"type":"text","text":" (no exponents on "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" other than 1)"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For part (a), substitute "},{"type":"inlineMath","attrs":{"latex":"x = -2"}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":"h(x)"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"h(-2) = -(-2)^2 + 5(-2) + 12"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Carefully compute "},{"type":"inlineMath","attrs":{"latex":"(-2)^2"}},{"type":"text","text":", then apply the negative sign and continue with the arithmetic."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Repeat the process for "},{"type":"inlineMath","attrs":{"latex":"x = 3"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"h(3) = -(3)^2 + 5 \\times 3 + 12"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For part (b), examine the structure of "},{"type":"inlineMath","attrs":{"latex":"h(x)"}},{"type":"text","text":". Does it fit the form "},{"type":"inlineMath","attrs":{"latex":"mx + b"}},{"type":"text","text":" (linear), or does it have a higher power of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"?"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Explain how the presence of "},{"type":"inlineMath","attrs":{"latex":"x^2"}},{"type":"text","text":" affects the function's classification."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) "},{"type":"inlineMath","attrs":{"latex":"h(-2) = -4 - 10 + 12 = -2"}},{"type":"text","text":"; "},{"type":"inlineMath","attrs":{"latex":"h(3) = -9 + 15 + 12 = 18"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) "},{"type":"inlineMath","attrs":{"latex":"h(x)"}},{"type":"text","text":" is non-linear because it contains an "},{"type":"inlineMath","attrs":{"latex":"x^2"}},{"type":"text","text":" term, making it a quadratic function."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Translating Verbal Phrases to Expressions"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Translating word phrases into algebraic expressions."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to convert verbal descriptions into mathematical expressions using variables and operations."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sum: addition"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quotient: division"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Less than: subtraction (order matters)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Cost expressions: multiplication and addition"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (a), identify the sum: "},{"type":"inlineMath","attrs":{"latex":"n + 8"}},{"type":"text","text":", then multiply by 3, then subtract 4."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (b), \"seven more than a number "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"\" is "},{"type":"inlineMath","attrs":{"latex":"y + 7"}},{"type":"text","text":", then divide by 10, then subtract 6."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (c), multiply the number of adult tickets "},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":" by 14, child tickets "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":" by 8, then add the two products."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) "},{"type":"inlineMath","attrs":{"latex":"3(n + 8) - 4"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) "},{"type":"inlineMath","attrs":{"latex":"\\frac{y + 7}{10} - 6"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"c) "},{"type":"inlineMath","attrs":{"latex":"C = 14a + 8c"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. Determining if a Relation is a Function"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Definition of a function (mapping from inputs to outputs)."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question asks you to determine if a given relation (table of values) is a function by checking if each input maps to only one output."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Function: Each input (x) must correspond to exactly one output (y)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Examine the table and look for repeated input values."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check if any input value is associated with more than one output value."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If so, explain why this violates the definition of a function."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The relation is "},{"type":"text","marks":[{"type":"bold"}],"text":"not"},{"type":"text","text":" a function because the input "},{"type":"inlineMath","attrs":{"latex":"x = -3"}},{"type":"text","text":" maps to two different outputs ("},{"type":"inlineMath","attrs":{"latex":"y = 8"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"y = 15"}},{"type":"text","text":")."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5. Writing and Solving Equations in Context"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Writing and solving linear equations from word problems."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to model a real-world scenario with a linear equation and solve for an unknown variable."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Linear equation: "},{"type":"inlineMath","attrs":{"latex":"y = mx + b"}},{"type":"text","text":" or similar forms"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Flat fee: constant term"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Hourly rate: coefficient of variable"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (a), write an equation: total cost = flat fee + (hourly rate) "},{"type":"inlineMath","attrs":{"latex":"\\times"}},{"type":"text","text":" hours."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set the total cost equal to $170"},{"type":"inlineMath","attrs":{"latex":" and let "}},{"type":"text","text":"h$ be the number of hours."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (b), isolate "},{"type":"inlineMath","attrs":{"latex":"h"}},{"type":"text","text":" by subtracting the flat fee from both sides."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by the hourly rate to solve for "},{"type":"inlineMath","attrs":{"latex":"h"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) "},{"type":"inlineMath","attrs":{"latex":"45 + 25h = 170"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) "},{"type":"inlineMath","attrs":{"latex":"h = 5"}},{"type":"text","text":" hours"}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. Rearranging Literal Equations"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Solving literal equations for a specified variable."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to manipulate equations with multiple variables to isolate a particular variable."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Literal equation: an equation involving two or more variables"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Isolate: get the specified variable alone on one side of the equation"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (a), factor "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" from the right side if possible, then divide both sides by the remaining factor to solve for $x$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For (b), multiply both sides by "},{"type":"inlineMath","attrs":{"latex":"2h"}},{"type":"text","text":" to clear the denominator, then isolate "},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":" by subtracting "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":" from both sides."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{P}{4y - 7z}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) "},{"type":"inlineMath","attrs":{"latex":"b = 2Ah - c"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7. Average Rate of Change from Tabular Data"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Average rate of change (AROC) from data tables."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to compute the change in a quantity over an interval and to find the average rate of change using tabular data."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Change: "},{"type":"inlineMath","attrs":{"latex":"\\Delta y = y_2 - y_1"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Average Rate of Change: "},{"type":"inlineMath","attrs":{"latex":"\\text{AROC} = \\frac{y_2 - y_1}{x_2 - x_1}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For the first 8 hours, subtract the population at "},{"type":"inlineMath","attrs":{"latex":"t = 0"}},{"type":"text","text":" from the population at "},{"type":"inlineMath","attrs":{"latex":"t = 8"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For the interval "},{"type":"inlineMath","attrs":{"latex":"t = 8"}},{"type":"text","text":" to "},{"type":"inlineMath","attrs":{"latex":"t = 20"}},{"type":"text","text":", use the AROC formula: "},{"type":"inlineMath","attrs":{"latex":"\\frac{P(20) - P(8)}{20 - 8}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Plug in the values from the table for "},{"type":"inlineMath","attrs":{"latex":"P(8)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"P(20)"}},{"type":"text","text":", and compute the numerator and denominator."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) Change in first 8 hours: "},{"type":"inlineMath","attrs":{"latex":"270 - 120 = 150"}},{"type":"text","text":" bacteria"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) AROC from "},{"type":"inlineMath","attrs":{"latex":"t=8"}},{"type":"text","text":" to "},{"type":"inlineMath","attrs":{"latex":"t=20"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"\\frac{930 - 270}{20 - 8} = \\frac{660}{12} = 55"}},{"type":"text","text":" bacteria per hour"}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8. Average Rate of Change over Intervals"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Average rate of change for a function over a closed interval."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to evaluate a function at two points and use those values to compute the average rate of change."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Average Rate of Change: "},{"type":"inlineMath","attrs":{"latex":"\\text{AROC} = \\frac{f(b) - f(a)}{b - a}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Function: "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\sqrt[3]{x} - 1"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluate "},{"type":"inlineMath","attrs":{"latex":"f(-8)"}},{"type":"text","text":": Compute "},{"type":"inlineMath","attrs":{"latex":"\\sqrt[3]{-8}"}},{"type":"text","text":", then subtract 1."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluate "},{"type":"inlineMath","attrs":{"latex":"f(1)"}},{"type":"text","text":": Compute "},{"type":"inlineMath","attrs":{"latex":"\\sqrt[3]{1}"}},{"type":"text","text":", then subtract 1."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the AROC formula with "},{"type":"inlineMath","attrs":{"latex":"a = -8"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"b = 1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation "},{"type":"inlineMath","attrs":{"latex":"\\frac{f(1) - f(-8)}{1 - (-8)}"}},{"type":"text","text":" and simplify as much as possible."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"AROC = "},{"type":"inlineMath","attrs":{"latex":"\\frac{0 - (-3)}{1 - (-8)} = \\frac{3}{9} = \\frac{1}{3}"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9. Direct Proportionality Functions"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Direct proportionality and finding the constant of proportionality."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to model a direct proportional relationship and use it to make predictions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Direct proportionality: "},{"type":"inlineMath","attrs":{"latex":"M = kL"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Constant of proportionality: "},{"type":"inlineMath","attrs":{"latex":"k = \\frac{M}{L}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express "},{"type":"inlineMath","attrs":{"latex":"M"}},{"type":"text","text":" as a function of "},{"type":"inlineMath","attrs":{"latex":"L"}},{"type":"text","text":" using "},{"type":"inlineMath","attrs":{"latex":"M = kL"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the given values ("},{"type":"inlineMath","attrs":{"latex":"M = 12"}},{"type":"text","text":" grams, "},{"type":"inlineMath","attrs":{"latex":"L = 30"}},{"type":"text","text":" cm) to solve for "},{"type":"inlineMath","attrs":{"latex":"k"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"State the units for "},{"type":"inlineMath","attrs":{"latex":"k"}},{"type":"text","text":" (grams per centimeter)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the function to find the mass when "},{"type":"inlineMath","attrs":{"latex":"L = 45"}},{"type":"text","text":" cm: "},{"type":"inlineMath","attrs":{"latex":"M(45) = k \\times 45"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) "},{"type":"inlineMath","attrs":{"latex":"M(L) = 0.4L"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) "},{"type":"inlineMath","attrs":{"latex":"k = 0.4"}},{"type":"text","text":" grams/cm"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"c) "},{"type":"inlineMath","attrs":{"latex":"M(45) = 0.4 \\times 45 = 18"}},{"type":"text","text":" grams"}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q10. Linear vs. Proportional Relationships"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Identifying direct proportionality in algebraic expressions."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to distinguish between direct proportionality and other types of relationships."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Direct proportionality: "},{"type":"inlineMath","attrs":{"latex":"y = kx"}},{"type":"text","text":" (must pass through the origin)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Linear function: "},{"type":"inlineMath","attrs":{"latex":"y = mx + b"}},{"type":"text","text":" (may have a nonzero intercept)"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each expression, check if it fits the form "},{"type":"inlineMath","attrs":{"latex":"y = kx"}},{"type":"text","text":" with "},{"type":"inlineMath","attrs":{"latex":"b = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If yes, identify "},{"type":"inlineMath","attrs":{"latex":"k"}},{"type":"text","text":"; if not, explain why (e.g., nonzero intercept, radical, etc.)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) Yes, "},{"type":"inlineMath","attrs":{"latex":"k = \\frac{1}{15}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) No, because it is a radical function, not linear."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"c) No, because it has a nonzero intercept ("},{"type":"inlineMath","attrs":{"latex":"+6"}},{"type":"text","text":")."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q11. Analyzing a Linear Function from a Graph"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Finding the equation of a line from a graph."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to determine the slope and y-intercept from a graph and write the equation in function notation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Slope formula: "},{"type":"inlineMath","attrs":{"latex":"m = \\frac{y_2 - y_1}{x_2 - x_1}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Function notation: "},{"type":"inlineMath","attrs":{"latex":"f(x) = mx + b"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the y-intercept by finding where the line crosses the y-axis ("},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Choose two clear integer points on the line and use them to calculate the slope "},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the equation in the form "},{"type":"inlineMath","attrs":{"latex":"f(x) = mx + b"}},{"type":"text","text":" using your values for "},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"f(x) = -3x + 2"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q12. Writing Linear Functions from Two Points"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Writing the equation of a line given two points."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the slope and use point-slope or slope-intercept form to write a linear function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Slope: "},{"type":"inlineMath","attrs":{"latex":"m = \\frac{y_2 - y_1}{x_2 - x_1}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Point-slope form: "},{"type":"inlineMath","attrs":{"latex":"y - y_1 = m(x - x_1)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the two points: "},{"type":"inlineMath","attrs":{"latex":"(-3, 17)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"(5, -7)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the slope "},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":" using the formula above."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use point-slope form with one of the points to write the equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rearrange to slope-intercept form if needed."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"y + 7 = -3(x - 5)"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"y = -3x + 8"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q13. Writing Linear Functions from Two Points"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Writing the equation of a line given two points."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question is similar to Q12, testing your ability to find the slope and write the equation using point-slope or slope-intercept form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Slope: "},{"type":"inlineMath","attrs":{"latex":"m = \\frac{y_2 - y_1}{x_2 - x_1}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Point-slope form: "},{"type":"inlineMath","attrs":{"latex":"y - y_1 = m(x - x_1)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the two points: "},{"type":"inlineMath","attrs":{"latex":"(4, -5)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"(12, -1)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the slope "},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use point-slope form with one of the points."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rearrange to slope-intercept form if needed."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"y + 5 = \\frac{1}{2}(x - 4)"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"y = \\frac{1}{2}x - 7"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q14. Comparing Linear Cost Models"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Setting up and solving equations to compare two linear models."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to model costs with linear equations, set them equal, and solve for the break-even point."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Linear cost model: "},{"type":"inlineMath","attrs":{"latex":"C(x) = mx + b"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Break-even: set two cost expressions equal and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the cost expression for each plan: Plan A includes a fixed fee and a per-class fee; Plan B is per-class only."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set the two expressions equal to find the number of classes where costs are the same."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" by isolating it on one side."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute the value of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" back into either cost expression to find the total cost at the break-even point."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) Plan A: "},{"type":"inlineMath","attrs":{"latex":"7x + 60"}},{"type":"text","text":"; Plan B: "},{"type":"inlineMath","attrs":{"latex":"13x"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) "},{"type":"inlineMath","attrs":{"latex":"x = 10"}},{"type":"text","text":" classes"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"c) Total cost at break-even: $130$"}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q15. Modeling with Linear Equations (Budget Constraint)"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Writing and solving a linear equation to model a budget constraint."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to set up an equation based on a real-world scenario and solve for the maximum number of days within a budget."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Linear equation: "},{"type":"inlineMath","attrs":{"latex":"\\text{Total cost} = \\text{fixed cost} + \\text{variable cost} \\times d"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the equation: "},{"type":"inlineMath","attrs":{"latex":"350 + 220d = 2400"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract the fixed cost ($350$) from both sides to isolate the variable term."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by the daily cost ($220"},{"type":"inlineMath","attrs":{"latex":") to solve for "}},{"type":"text","text":"d$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Interpret the result in terms of whole days (since partial days may not be possible)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a) "},{"type":"inlineMath","attrs":{"latex":"350 + 220d = 2400"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b) Maximum number of whole days: $9$"}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q16. Solving Multi-Step Inequalities"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Solving linear inequalities with multiple steps."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to apply the distributive property, combine like terms, and solve for the variable in an inequality."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Distributive property: "},{"type":"inlineMath","attrs":{"latex":"a(b + c) = ab + ac"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solving inequalities: same as equations, but reverse the inequality if multiplying/dividing by a negative"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Distribute "},{"type":"inlineMath","attrs":{"latex":"-5"}},{"type":"text","text":" across "},{"type":"inlineMath","attrs":{"latex":"(2x - 3)"}},{"type":"text","text":" and combine with "},{"type":"inlineMath","attrs":{"latex":"+8"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Move all terms involving "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to one side and constants to the other."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":", remembering to reverse the inequality if necessary."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Graph the solution on a number line (sketch in your workspace)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x \\geq 6"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q17. Writing Equations of Parallel Lines"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Writing the equation of a line parallel to a given line through a specific point."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the slope of a given line and use point-slope form to write the equation of a parallel line."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Parallel lines: same slope"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Point-slope form: "},{"type":"inlineMath","attrs":{"latex":"y - y_1 = m(x - x_1)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite "},{"type":"inlineMath","attrs":{"latex":"3x + 4y = 12"}},{"type":"text","text":" in slope-intercept form to find the slope "},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the same slope for the parallel line."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Plug the given point "},{"type":"inlineMath","attrs":{"latex":"(-8, 5)"}},{"type":"text","text":" and the slope into point-slope form."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify to the desired form if needed."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"y - 5 = -\\frac{3}{4}(x + 8)"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q18. Writing Equations of Perpendicular Lines"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Writing the equation of a line perpendicular to a given line through a specific point."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the negative reciprocal of a slope and use point-slope form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Perpendicular lines: slopes are negative reciprocals"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Point-slope form: "},{"type":"inlineMath","attrs":{"latex":"y - y_1 = m(x - x_1)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the slope of the given line "},{"type":"inlineMath","attrs":{"latex":"y = -3x + 7"}},{"type":"text","text":" ("},{"type":"inlineMath","attrs":{"latex":"m = -3"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the negative reciprocal: "},{"type":"inlineMath","attrs":{"latex":"m_{\\perp} = \\frac{1}{3}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use point-slope form with point "},{"type":"inlineMath","attrs":{"latex":"(6, -2)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"m_{\\perp}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify to the desired form if needed."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"y + 2 = \\frac{1}{3}(x - 6)"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q19. Solving Systems of Equations by Elimination"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Solving systems of linear equations using the elimination method."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to manipulate two equations to eliminate one variable and solve for the other."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Elimination method: Multiply equations as needed to align coefficients, then add/subtract to eliminate a variable."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the system: "},{"type":"inlineMath","attrs":{"latex":"4x - 3y = 23"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"3x + 2y = 13"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply each equation by a suitable number so that the coefficients of "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" (or "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":") are opposites."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add or subtract the equations to eliminate one variable."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for the remaining variable, then substitute back to find the other variable."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"(x, y) = (5, -1)"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q20. Special Solution Systems (No/Infinite Solutions)"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Determining the type of solution for a system of equations."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to recognize when a system has one, none, or infinitely many solutions, and to explain why."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Consistent and dependent: infinitely many solutions"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Consistent and independent: one solution"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Inconsistent: no solution"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the system: "},{"type":"inlineMath","attrs":{"latex":"12x - 8y = -20"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"-3x + 2y = 5"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the second equation by a number to align coefficients with the first equation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add the equations and observe if you get a true statement (like "},{"type":"inlineMath","attrs":{"latex":"0 = 0"}},{"type":"text","text":") or a contradiction."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Explain what this result means about the system's solutions."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Infinitely many solutions; the two equations represent the same line."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q21. Real-World Systems Applications: Sales"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Setting up and solving a system of equations from a word problem."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to define variables, write equations for totals and revenue, and solve for unknowns."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Let "},{"type":"inlineMath","attrs":{"latex":"s"}},{"type":"text","text":" = number of standard bagels, "},{"type":"inlineMath","attrs":{"latex":"p"}},{"type":"text","text":" = number of premium bagels"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total bagels: "},{"type":"inlineMath","attrs":{"latex":"s + p = 140"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total revenue: "},{"type":"inlineMath","attrs":{"latex":"1.50s + 2.50p = 270"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Define your variables clearly."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the two equations based on the problem statement."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use substitution or elimination to solve for one variable."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute back to find the other variable."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"80 standard bagels, 60 premium bagels"}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q22. Real-World Systems Applications: Finance"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"b