Quiz: Linear Inequalities and Absolute Value
Test your understanding of linear inequalities, compound inequalities, and absolute value equations with these questions.
1. When solving -3x > 12, what must you remember to do?
- Add 3 to both sides
- Reverse the inequality symbol when dividing by -3
- Change > to ≥
- Multiply both sides by -3
Show Answer
The correct answer is B. When dividing or multiplying both sides of an inequality by a negative number, you must reverse the inequality symbol. So -3x > 12 becomes x < -4 after dividing by -3. This is a crucial rule that distinguishes inequality solving from equation solving. Option A doesn't address the coefficient. Option C changes the wrong aspect. Option D would make the inequality worse.
Concept Tested: Solving Linear Inequalities
2. Which graph correctly represents x ≤ 3?
- Open circle at 3, shade right
- Open circle at 3, shade left
- Closed circle at 3, shade left
- Closed circle at 3, shade right
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The correct answer is C. The symbol ≤ means "less than or equal to," so we use a closed circle at 3 (because 3 is included in the solution) and shade to the left (for all numbers less than 3). An open circle is used for < or >, while a closed circle is used for ≤ or ≥. The shading direction indicates whether we want numbers greater than (shade right) or less than (shade left) the boundary point.
Concept Tested: Graphing Inequalities
3. Solve the compound inequality -2 < x + 1 < 5. What is the solution?
- -3 < x < 4
- -1 < x < 6
- -3 < x < 6
- -1 < x < 4
Show Answer
The correct answer is A. Subtract 1 from all three parts of the inequality: -2 - 1 < x + 1 - 1 < 5 - 1, which gives -3 < x < 4. This means x is between -3 and 4 (not including the endpoints). Option B incorrectly adds 1 instead of subtracting. Options C and D mix errors from both operations.
Concept Tested: And Inequalities
See: And Inequalities
4. What are the solutions to |x| = 8?
- x = 8 only
- x = -8 only
- No solution
- x = 8 or x = -8
Show Answer
The correct answer is D. The absolute value |x| = 8 means x is 8 units from zero on the number line. Both 8 and -8 are exactly 8 units from zero, so both are solutions. Absolute value equations typically have two solutions (except when the right side is 0, which gives one solution). Options A and B give only one of the two solutions. Option C would apply if the equation were |x| = -8.
Concept Tested: Absolute Value Equations
5. Solve |x - 4| < 6. Which is the correct solution?
- -2 < x < 10
- x < -2 or x > 10
- -10 < x < 2
- x < -10 or x > 2
Show Answer
The correct answer is A. For |x - 4| < 6, this means x - 4 is less than 6 units from zero, so -6 < x - 4 < 6. Adding 4 to all parts gives -2 < x < 10. The "less than" type of absolute value inequality becomes an "and" compound inequality. Option B incorrectly treats it as an "or" inequality (which would be for >). Options C and D have incorrect arithmetic.
Concept Tested: Absolute Value Inequalities
6. Which compound inequality represents "x < -1 OR x ≥ 4"?
- An "and" inequality with solution between -1 and 4
- An "or" inequality with two separate regions
- A single inequality x ≥ 3
- An absolute value inequality
Show Answer
The correct answer is B. The word "OR" indicates an "or" compound inequality, meaning the solution includes all numbers less than -1 as well as all numbers greater than or equal to 4. This creates two separate solution regions on the number line, with a gap between them. Option A describes an "and" inequality. Option C attempts to combine them incorrectly. Option D is a different type of problem.
Concept Tested: Or Inequalities
See: Or Inequalities
7. What is the solution to |2x + 5| > 9?
- -7 < x < 2
- x < -7 or x > 2
- -2 < x < 7
- x < -2 or x > 7
Show Answer
The correct answer is B. For |2x + 5| > 9, we set up: 2x + 5 < -9 OR 2x + 5 > 9. Solving the first: 2x < -14, so x < -7. Solving the second: 2x > 4, so x > 2. The solution is x < -7 or x > 2. The "greater than" type of absolute value inequality becomes an "or" compound inequality. Option A would be for the "less than" type. Options C and D have calculation errors.
Concept Tested: Absolute Value Inequalities
8. A temperature must stay between 35°F and 40°F. Which inequality represents this?
- T < 35 or T > 40
- 35 < T < 40
- |T - 37.5| < 2.5
- Both B and C are correct
Show Answer
The correct answer is D. The temperature T must satisfy 35 < T < 40 (or using ≤ if endpoints are included). This can also be expressed as an absolute value inequality: |T - 37.5| < 2.5, where 37.5 is the midpoint and 2.5 is the tolerance. Both representations describe the same range. Option A describes temperatures outside the range. Options B and C are each correct individually, making D the best answer.
Concept Tested: Applications of Linear Equations and Inequalities
See: Applications
9. When solving 2x + 3 ≤ 11, what is the solution set in interval notation?
- [4, ∞)
- (-∞, 4]
- (4, ∞)
- (-∞, 4)
Show Answer
The correct answer is B. Subtracting 3 gives 2x ≤ 8, then dividing by 2 gives x ≤ 4. In interval notation, this is (-∞, 4], where the bracket at 4 indicates that 4 is included (because of ≤), and (-∞ indicates all numbers less than 4. Option A reverses the direction. Option D uses a parenthesis instead of a bracket, which would indicate x < 4 instead of x ≤ 4.
Concept Tested: Linear Inequality / Solving Linear Inequalities
10. What happens when you try to solve |x + 2| = -5?
- x = -7 or x = 3
- x = 3 or x = -7
- x = -2
- No solution (absolute value cannot be negative)
Show Answer
The correct answer is D. An absolute value represents distance from zero, which is always non-negative. Therefore, |x + 2| cannot equal -5, a negative number. This equation has no solution. Equations like |x| = a where a < 0 always have no solution. Options A and B attempt to solve as if the right side were positive 5. Option C incorrectly assumes one solution.
Concept Tested: Absolute Value Equations
See: Special Cases