Unit 2 Review A – Beginning Algebra Study Guidance

Q1. Insert <, >, or = to make a true statement: −6.3 ____ −6.03

Background

Topic: Comparing Real Numbers

This question tests your understanding of how to compare negative decimal numbers and determine which is greater, less, or equal.

Key Terms:

• Negative numbers: Numbers less than zero.

• Decimal numbers: Numbers with a fractional part separated by a decimal point.

Step-by-Step Guidance

1. Recall that for negative numbers, the number with the smaller absolute value is actually greater.

2. Compare the absolute values: $|-6.3| = 6.3$ and $|-6.03| = 6.03$.

3. Think about which number is further from zero on the number line.

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Q2. Insert <, >, or = to make a true statement: 5 ____ |−3|

Background

Topic: Absolute Value and Comparing Numbers

This question tests your understanding of absolute value and how to compare it to another number.

Key Terms:

• Absolute value: The distance of a number from zero, always positive.

Step-by-Step Guidance

1. Calculate the absolute value: $| -3 | = 3$.

2. Compare 5 and 3.

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Q3. Solve $5x - 2y = 4$ for $y$. Write the answer in $y = mx + b$ form.

Background

Topic: Solving Linear Equations for a Variable

This question tests your ability to rearrange a linear equation to solve for $y$ and express it in slope-intercept form.

Key Terms and Formula:

• Slope-intercept form: $y = mx + b$

• Linear equation: An equation involving variables to the first power.

Step-by-Step Guidance

1. Start with the equation: $5x - 2y = 4$.

2. Isolate the $y$ term by subtracting $5x$ from both sides: $-2y = 4 - 5x$.

3. Divide both sides by $-2$ to solve for $y$.

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Q4. Solve for $F$ in $C = \frac{5}{9}(F - 32)$.

Background

Topic: Solving Formulas for a Variable

This question tests your ability to manipulate a formula to solve for a specific variable.

Key Terms and Formula:

• Formula: $C = \frac{5}{9}(F - 32)$

• Solving for $F$ means expressing $F$ in terms of $C$.

Step-by-Step Guidance

1. Start with $C = \frac{5}{9}(F - 32)$.

2. Multiply both sides by 9 to eliminate the denominator: $9C = 5(F - 32)$.

3. Divide both sides by 5: $\frac{9C}{5} = F - 32$.

4. Add 32 to both sides to solve for $F$.

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Q5. Yong traveled to Florida for spring break at an average speed of 61.7 miles per hour for 5 hours. How far did he travel?

Background

Topic: Distance, Rate, and Time

This question tests your ability to use the formula relating distance, rate, and time.

Key Formula:

• $\text{Distance} = \text{Rate} \times \text{Time}$

Step-by-Step Guidance

1. Identify the known values: Rate $= 61.7$ mph, Time $= 5$ hours.

2. Set up the formula: $\text{Distance} = 61.7 \times 5$.

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Q6. The selling price of six homes shown by a real estate agent are $456,000, $420,000, $440,000, $410,000, $450,000, and $430,000. Find the median price of these six homes.

Background

Topic: Measures of Central Tendency

This question tests your ability to find the median of a set of numbers.

Key Terms:

• Median: The middle value when numbers are arranged in order. If there is an even number of values, the median is the average of the two middle values.

Step-by-Step Guidance

1. Arrange the prices in order from least to greatest.

2. Identify the two middle values since there are six prices.

3. Calculate the average of the two middle values.

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Q7. A mean of 90 is needed to make an A in a course. On his first 4 exams, Jeremy scored 92, 94, 88, and 98. What is the minimum score Jeremy can receive on the fifth exam to get an A in the course?

Background

Topic: Mean (Average) Calculation

This question tests your ability to calculate the required score to achieve a desired average.

Key Formula:

• $\text{Mean} = \frac{\text{Sum of scores}}{\text{Number of exams}}$

Step-by-Step Guidance

1. Let $x$ be the score needed on the fifth exam.

2. Set up the equation: $\frac{92 + 94 + 88 + 98 + x}{5} = 90$.

3. Multiply both sides by 5 to eliminate the denominator.

4. Solve for $x$.

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Q8. Mary invested a certain amount of money in a savings account paying 3% simple interest per year. When she withdrew her money at the end of 3 years, she received $450 in interest. How much did Mary place in the saving account?

Background

Topic: Simple Interest

This question tests your ability to use the simple interest formula to find the principal.

Key Formula:

• $I = P r t$

• $I$ = interest earned, $P$ = principal (amount invested), $r$ = rate (as a decimal), $t$ = time (years)

Step-by-Step Guidance

1. Identify the known values: $I = 450$, $r = 0.03$, $t = 3$.

2. Set up the formula: $450 = P \times 0.03 \times 3$.

3. Solve for $P$ by dividing both sides by $(0.03 \times 3)$.

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Q9. A yield traffic sign is triangular with a base of 36 inches and a height of 31 inches. Determine the area of the sign.

Background

Topic: Area of a Triangle

This question tests your ability to use the area formula for triangles.

Key Formula:

• $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Step-by-Step Guidance

1. Identify the base and height: base $= 36$ inches, height $= 31$ inches.

2. Set up the formula: $\text{Area} = \frac{1}{2} \times 36 \times 31$.

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Q10. Solve the inequality: $3x + 4 \geq 10$. Graph the solution on a number line and represent the solution in interval notation.

Background

Topic: Solving Linear Inequalities

This question tests your ability to solve inequalities and express the solution graphically and in interval notation.

Key Terms:

• Inequality: A mathematical statement that compares two expressions.

• Interval notation: A way to represent all numbers between two endpoints.

Step-by-Step Guidance

1. Start with $3x + 4 \geq 10$.

2. Subtract 4 from both sides: $3x \geq 6$.

3. Divide both sides by 3 to solve for $x$.

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Q11. Solve the inequality: $2(x + 3) > 6x - 4x + 4$. Graph the solution on a number line and represent the solution in interval notation.

Background

Topic: Solving Linear Inequalities

This question tests your ability to simplify and solve inequalities.

Key Terms:

• Combine like terms: Add or subtract terms with the same variable.

Step-by-Step Guidance

1. Expand the left side: $2(x + 3) = 2x + 6$.

2. Simplify the right side: $6x - 4x + 4 = 2x + 4$.

3. Set up the inequality: $2x + 6 > 2x + 4$.

4. Subtract $2x$ from both sides.

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Q12. Solve the inequality: $2(x + 4) \leq 2x - 5$. Graph the solution on a number line and represent the solution in interval notation.

Background

Topic: Solving Linear Inequalities

This question tests your ability to solve inequalities and represent the solution.

Key Terms:

• Distributive property: $a(b + c) = ab + ac$

Step-by-Step Guidance

1. Expand the left side: $2(x + 4) = 2x + 8$.

2. Set up the inequality: $2x + 8 \leq 2x - 5$.

3. Subtract $2x$ from both sides.

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Q13. Solve the inequality: $x - 8 \leq -3x + 11$. Graph the solution on a number line and represent the solution in interval notation.

Background

Topic: Solving Linear Inequalities

This question tests your ability to solve inequalities and represent the solution.

Key Terms:

• Move all terms with $x$ to one side.

Step-by-Step Guidance

1. Add $3x$ to both sides: $x + 3x - 8 \leq 11$.

2. Simplify: $4x - 8 \leq 11$.

3. Add 8 to both sides.

4. Solve for $x$.

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Q14. $\frac{x}{2} < \frac{2}{3}(x + 3)$

Background

Topic: Solving Linear Inequalities with Fractions

This question tests your ability to solve inequalities involving fractions.

Key Terms:

• Clear fractions by multiplying both sides by a common denominator.

Step-by-Step Guidance

1. Multiply both sides by 6 to clear denominators.

2. Simplify both sides after multiplying.

3. Solve for $x$.

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Q15. The number of packages of peppermint flavored candy sold was 105 packages less than 7 times the number of packages of orange flavored candy sold. Select a variable to represent one quantity and state what it represents. Express the second quantity in terms of the variable selected.

Background

Topic: Writing Expressions

This question tests your ability to use variables to represent quantities and write expressions.

Key Terms:

• Let $x$ represent the number of orange flavored candy packages sold.

Step-by-Step Guidance

1. Define $x$ as the number of orange packages sold.

2. Express the peppermint packages as $7x - 105$.

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Q16. The sum of two integers is 158. Determine the two integers if the larger is 10 less than twice the smaller.

Background

Topic: Systems of Equations

This question tests your ability to set up and solve a system of equations.

Key Terms:

• Let $x$ be the smaller integer, $y$ be the larger integer.

Step-by-Step Guidance

1. Set up the equations: $x + y = 158$ and $y = 2x - 10$.

2. Substitute $y$ from the second equation into the first.

3. Solve for $x$.

4. Find $y$ using the value of $x$.

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Q17. William is going to hire a snow plowing service. Elizabeth charges an annual fee of $80, plus $5 each time she plows. Jon charges an annual fee of $50, plus $10 each time he plows. How many times would the snow need to be plowed for the cost of both people to be the same?

Background

Topic: Linear Equations

This question tests your ability to set up and solve equations involving costs.

Key Terms:

• Let $x$ be the number of times plowed.

Step-by-Step Guidance

1. Write the cost equations: Elizabeth: $80 + 5x$, Jon: $50 + 10x$.

2. Set the two equations equal: $80 + 5x = 50 + 10x$.

3. Solve for $x$.

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Q18. A triangle has a perimeter of 75 inches. Determine the three sides if one side is 15 inches larger than the smallest side, and the third side is twice the smallest side.

Background

Topic: Systems of Equations

This question tests your ability to set up and solve equations based on relationships between sides.

Key Terms:

• Let $x$ be the smallest side.

Step-by-Step Guidance

1. Let the sides be $x$, $x + 15$, and $2x$.

2. Set up the equation: $x + (x + 15) + 2x = 75$.

3. Solve for $x$.

4. Find the other two sides using $x$.

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Q19. Alice and Bonnie start running at the same time from the same point and run in the same direction. Alice runs at 8 mph while Bonnie runs at a slower pace. After 2 hours they are 4 miles apart. Determine the speed at which Bonnie is running.

Background

Topic: Distance, Rate, and Time

This question tests your ability to set up and solve equations involving rates and distances.

Key Terms:

• Let $x$ be Bonnie's speed in mph.

Step-by-Step Guidance

1. Distance Alice: $8 \times 2 = 16$ miles.

2. Distance Bonnie: $x \times 2$ miles.

3. The difference in their distances is 4 miles: $16 - 2x = 4$.

4. Solve for $x$.

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Q20. How many liters of 20% salt solution must be added to 60 liters of 40% salt solution to get a solution that is 35% salt?

Background

Topic: Mixture Problems

This question tests your ability to set up and solve mixture equations.

Key Terms:

• Let $x$ be the liters of 20% solution to add.

Step-by-Step Guidance

1. Amount of salt in 20% solution: $0.20x$.

2. Amount of salt in 40% solution: $0.40 \times 60$.

3. Total volume: $x + 60$.

4. Set up the equation: $0.20x + 0.40 \times 60 = 0.35(x + 60)$.

5. Solve for $x$.

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Q21. Martha Goshaw wishes to place part of $12,000 into a saving account earning 8% simple interest and part into a saving account earning 7¼% simple interest. How much should she invest in each if she wishes to earn $900 in interest for the year?

Background

Topic: Simple Interest and Systems of Equations

This question tests your ability to set up and solve a system involving two accounts and total interest earned.

Key Terms:

• Let $x$ be the amount invested at 8%, $y$ at 7.25%.

Step-by-Step Guidance

1. Set up the equations: $x + y = 12,000$ and $0.08x + 0.0725y = 900$.

2. Solve the system using substitution or elimination.

Try solving on your own before revealing the answer!