Rise Placement Test Solving Equations and Inequalities Questions and Answers — Questions and Answers
Question 1: A student is saving for a trip that costs $540. They have already saved $120 and can save an additional $35 per week. Which inequality correctly represents the number of weeks, w, the student still needs to save to have at least enough money for the trip?
- 120 + 35w ≤ 540
- 120w + 35 ≥ 540
- 120 + 35w ≥ 540 (Correct answer)
- 120w + 35 ≤ 540
Correct answer: 120 + 35w ≥ 540
The total amount saved will be the initial $120 plus the amount saved per week ($35) times the number of weeks (w). This total must be 'at least' $540, which translates to 'greater than or equal to' (≥). Therefore, the correct inequality is 120 + 35w ≥ 540.
Question 2: Solve the following equation for x: 3(x + 2) - 4 = 2x + 9
- x = 5
- x = 7 (Correct answer)
- x = 1
- x = -1
Correct answer: x = 7
First, distribute the 3 on the left side: 3x + 6 - 4 = 2x + 9. Simplify the left side: 3x + 2 = 2x + 9. Next, subtract 2x from both sides: x + 2 = 9. Finally, subtract 2 from both sides to isolate x: x = 7.
Question 3: Which of the following is the correct solution set for the absolute value inequality |2x - 1| > 7?
- x < -3 or x > 4 (Correct answer)
- x < -4 or x > 3
- -3 < x < 4
- x > 4
Correct answer: x < -3 or x > 4
An absolute value inequality of the form |A| > B is solved by setting up two separate inequalities: A > B or A < -B. So, we solve 2x - 1 > 7 and 2x - 1 < -7. For the first, add 1 to both sides to get 2x > 8, then divide by 2 to get x > 4. For the second, add 1 to both sides to get 2x < -6, then divide by 2 to get x < -3. The solution set is x < -3 or x > 4.
Question 4: A taxi company charges a flat fee of $3.00 plus an additional $0.75 per mile. If a passenger has at most $15.00 to spend, what is the maximum number of miles they can travel?
- 20 miles
- 16 miles (Correct answer)
- 24 miles
- 15 miles
Correct answer: 16 miles
Let 'm' be the number of miles. The total cost is represented by the inequality 3.00 + 0.75m ≤ 15.00. To solve, first subtract 3.00 from both sides: 0.75m ≤ 12.00. Then, divide both sides by 0.75: m ≤ 16. The maximum number of miles is 16.
Question 5: What are the solutions to the quadratic equation x² - 3x - 10 = 0?
- x = 2 and x = -5
- x = -2 and x = -5
- x = 10 and x = -1
- x = 5 and x = -2 (Correct answer)
Correct answer: x = 5 and x = -2
This quadratic equation can be solved by factoring. We need to find two numbers that multiply to -10 and add to -3. These numbers are -5 and +2. So, the equation can be factored as (x - 5)(x + 2) = 0. Using the zero-product property, we set each factor to zero: x - 5 = 0 or x + 2 = 0. Solving these gives x = 5 and x = -2.
Question 6: Solve the inequality: -2(x - 4) < 10
- x < -1
- x > 1
- x > -1 (Correct answer)
- x < 1
Correct answer: x > -1
First, distribute the -2: -2x + 8 < 10. Next, subtract 8 from both sides: -2x < 2. Finally, divide both sides by -2. Remember to reverse the inequality sign when multiplying or dividing by a negative number: x > -1.
Rise Placement Test Solving Equations and Inequalities Questions and Answers
A student is saving for a trip that costs $540.
They have already saved $120 and can save an additional $35 per week.
Which inequality correctly represents the number of weeks, w, the student still needs to save to have at least enough money for the trip?