Q1a. Solve the equation:

Background

Topic: Linear Equations

This question tests your ability to solve linear equations with variables on both sides, including distributing and combining like terms.

Key Terms and Formulas:

• Distributive Property:

• Combining like terms

• Solving for a variable

Step-by-Step Guidance

1. Apply the distributive property to both sides: Expand and .

2. Combine like terms on each side of the equation.

3. Move all terms involving to one side and constants to the other.

4. Set up the equation to isolate .

Try solving on your own before revealing the answer!

Q1b. Solve the equation:

Background

Topic: Linear Equations with Fractions

This question tests your ability to solve linear equations involving fractions and variables on both sides.

Key Terms and Formulas:

• Multiplying both sides by the least common denominator (LCD) to clear fractions

• Solving for a variable

Step-by-Step Guidance

1. Expand using the distributive property.

2. Find the LCD for 7 and 5, then multiply both sides by the LCD to clear fractions.

3. Combine like terms and isolate .

4. Set up the equation for the final calculation.

Try solving on your own before revealing the answer!

Q1c. Given , solve for .

Background

Topic: Solving Formulas for a Variable

This question tests your ability to manipulate formulas to isolate a specific variable.

Key Terms and Formulas:

• Solving for a variable in terms of others

Step-by-Step Guidance

1. Start with .

2. Subtract from both sides to isolate terms with .

3. Divide both sides by 2 to solve for .

Try solving on your own before revealing the answer!

Q2a. Solve the compound inequality:

Background

Topic: Compound Inequalities

This question tests your ability to solve compound inequalities and express the solution in set-builder notation, interval notation, and on a number line.

Key Terms and Formulas:

• Compound inequality

• Set-builder notation

• Interval notation

Step-by-Step Guidance

1. Break the compound inequality into two parts: and .

2. Solve each part for by isolating .

3. Combine the solution sets to find the range of .

4. Set up the solution in set-builder and interval notation.

Try solving on your own before revealing the answer!

Q2b. Solve the inequality: or

Background

Topic: Compound (OR) Inequalities

This question tests your ability to solve inequalities connected by "or" and express the solution in multiple forms.

Key Terms and Formulas:

• "Or" compound inequality

• Set-builder notation

• Interval notation

Step-by-Step Guidance

1. Solve for .

2. Solve for .

3. Combine the solution sets using "or".

4. Set up the solution in set-builder and interval notation.

Try solving on your own before revealing the answer!

Q3a. Three subtracted from five times a number is thirty-seven. What is the number?

Background

Topic: Translating Words to Equations

This question tests your ability to convert a verbal statement into an algebraic equation and solve for the unknown.

Key Terms and Formulas:

• Let the unknown be

• "Five times a number" is

• "Three subtracted from" means

Step-by-Step Guidance

1. Write the equation:

2. Add 3 to both sides to isolate .

3. Divide both sides by 5 to solve for .

Try solving on your own before revealing the answer!

Q3b. Three times the sum of a number and four is negative six. What is the number?

Background

Topic: Translating Words to Equations

This question tests your ability to convert a verbal statement into an algebraic equation and solve for the unknown.

Key Terms and Formulas:

• Let the unknown be

• "The sum of a number and four" is

• "Three times the sum" is

Step-by-Step Guidance

1. Write the equation:

2. Expand using the distributive property.

3. Isolate by subtracting and dividing as needed.

Try solving on your own before revealing the answer!

Q4. A sheet of metal with an area of 30 square feet will be formed into a box 6 feet long by 1.5 feet wide. Use the formula for surface area: to find the height of the box.

Background

Topic: Solving Formulas for a Variable

This question tests your ability to use a formula for surface area and solve for an unknown dimension.

Key Terms and Formulas:

• Surface area formula for a rectangular box:

• Given: , ,

Step-by-Step Guidance

1. Plug the known values into the formula:

2. Calculate and simplify the equation.

3. Combine like terms involving .

4. Set up the equation to solve for .

Try solving on your own before revealing the answer!

Q5a. Graph the coordinate pairs (3, 5), (-2, 4), (1, -3), and (-5, -4)

Background

Topic: Plotting Points on the Cartesian Plane

This question tests your ability to plot points on a coordinate grid.

Key Terms and Formulas:

• Cartesian plane

• Ordered pairs

Step-by-Step Guidance

1. Identify each coordinate pair and locate it on the grid.

2. Plot each point accurately using the and values.

3. Check that all points are placed correctly relative to the axes.

Try plotting on your own before revealing the answer!

Q5b. Graph the line . State the slope, the y-intercept, and the coordinates of at least TWO additional points on this line.

Background

Topic: Linear Equations and Graphing

This question tests your ability to graph a line, identify its slope and y-intercept, and find additional points.

Key Terms and Formulas:

• Slope-intercept form:

• Slope (): the rate of change

• Y-intercept (): where the line crosses the -axis

Step-by-Step Guidance

1. Identify the slope () and y-intercept ().

2. Plot the y-intercept at .

3. Use the slope to find another point: from , move up 2 units and right 1 unit to .

4. Choose another value (e.g., ) and calculate to find a second point.

Try graphing and finding points before revealing the answer!

Q5c. Graph the line . State the slope, the y-intercept, and the coordinates of at least TWO additional points on this line.

Background

Topic: Linear Equations and Graphing

This question tests your ability to graph a line with a fractional slope and negative y-intercept.

Key Terms and Formulas:

• Slope-intercept form:

• Slope ()

• Y-intercept ()

Step-by-Step Guidance

1. Identify the slope and y-intercept.

2. Plot the y-intercept at .

3. Use the slope to find another point: from , move down 4 units and right 5 units to .

4. Choose another value (e.g., ) and calculate to find a second point.

Try graphing and finding points before revealing the answer!

Q5d. Graph the line . State the slope, the y-intercept, and the coordinates of at least TWO additional points on this line.

Background

Topic: Linear Equations and Graphing

This question tests your ability to graph a line with a fractional slope and positive y-intercept.

Key Terms and Formulas:

• Slope-intercept form:

• Slope ()

• Y-intercept ()

Step-by-Step Guidance

1. Identify the slope and y-intercept.

2. Plot the y-intercept at .

3. Use the slope to find another point: from , move up 2 units and right 3 units to .

4. Choose another value (e.g., ) and calculate to find a second point.

Try graphing and finding points before revealing the answer!

Q6a. Find the point-slope equation for a line with slope 3 that passes through the point (3, -5)

Background

Topic: Point-Slope Form of a Line

This question tests your ability to write the equation of a line given a slope and a point.

Key Terms and Formulas:

• Point-slope form:

• Slope (), point

Step-by-Step Guidance

1. Identify , .

2. Plug these values into the point-slope formula.

3. Write the equation in point-slope form.

Try writing the equation before revealing the answer!

Q6b. Find the slope of the line that passes through points (1, 3) and (-2, -9)

Background

Topic: Slope Formula

This question tests your ability to calculate the slope between two points.

Key Terms and Formulas:

• Slope formula:

Step-by-Step Guidance

1. Label the points: , .

2. Plug the values into the slope formula.

3. Calculate the numerator and denominator separately.

Try calculating the slope before revealing the answer!

Q6c. Find the slope-intercept equation of the line parallel to that passes through the point (2, -1)

Background

Topic: Parallel Lines and Slope-Intercept Form

This question tests your ability to write the equation of a line parallel to a given line, passing through a specific point.

Key Terms and Formulas:

• Parallel lines have the same slope

• Slope-intercept form:

Step-by-Step Guidance

1. Identify the slope of the given line ().

2. Use the point and the slope to write the equation in point-slope form.

3. Convert to slope-intercept form by solving for .

Try writing the equation before revealing the answer!

Q6d. Find the slope-intercept equation of the line perpendicular to that passes through the point (-1, 5)

Background

Topic: Perpendicular Lines and Slope-Intercept Form

This question tests your ability to write the equation of a line perpendicular to a given line, passing through a specific point.

Key Terms and Formulas:

• Perpendicular lines: slopes are negative reciprocals

• Slope-intercept form:

Step-by-Step Guidance

1. Find the negative reciprocal of the given slope ().

2. Use the point (-1, 5) and the slope to write the equation in point-slope form.

3. Convert to slope-intercept form by solving for .

Try writing the equation before revealing the answer!

Q7a. Graph the inequality (shade the proper region)

Background

Topic: Graphing Linear Inequalities

This question tests your ability to graph a linear inequality and shade the correct region.

Key Terms and Formulas:

• Linear inequality

• Shading above or below the line

Step-by-Step Guidance

1. Graph the boundary line (solid line because of ).

2. Determine which side to shade (above the line for ).

3. Test a point (e.g., (0,0)) to confirm the region.

Try graphing and shading before revealing the answer!

Q7b. Graph the inequality (shade the proper region)

Background

Topic: Graphing Linear Inequalities

This question tests your ability to graph a linear inequality and shade the correct region.

Key Terms and Formulas:

• Linear inequality

• Shading above or below the line

Step-by-Step Guidance

1. Rewrite as for easier graphing.

2. Graph the boundary line (dashed line because of ).

3. Shade the region above the line.

Try graphing and shading before revealing the answer!

Q8a. Determine if the relation {(3, 3), (5, 3), (7, 3)} is a function

Background

Topic: Functions and Relations

This question tests your ability to determine if a relation is a function based on the definition.

Key Terms and Formulas:

• Function: Each input () has only one output ()

Step-by-Step Guidance

1. Check if any value is repeated with different values.

2. Determine if the relation meets the definition of a function.

Try deciding before revealing the answer!

Q8b. Determine if the relation {(2, 2), (-18, 5), (3, -6), (2, 15), (-9, 3)} is a function

Background

Topic: Functions and Relations

This question tests your ability to determine if a relation is a function based on the definition.

Key Terms and Formulas:

• Function: Each input () has only one output ()

Step-by-Step Guidance

1. Check if any value is repeated with different values.

2. Determine if the relation meets the definition of a function.

Try deciding before revealing the answer!

Q9. Consider the following functions and find the evaluations listed below. Solutions must be in simplest form.

Background

Topic: Function Evaluation

This question tests your ability to evaluate functions for given values of .

Key Terms and Formulas:

• Function notation: , ,

• Substitute the given value for

Step-by-Step Guidance

1. For each function, substitute the given value of into the formula.

2. Perform the arithmetic operations as indicated.

3. Simplify the result to its simplest form.

Try evaluating before revealing the answer!