Algebra 1 Practice Test: Absolute Value Equations and Inequalities
- Absolute Value Equations and Inequalities
- 10 questions
- 13 minutes
These 10 questions cover equations like |ax + b| = c and inequalities with less than or greater than. Always isolate the absolute value first, and remember that it can never equal a negative number.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
Solve |x − 3| = 7.
Correct answer: C
Split into x − 3 = 7 and x − 3 = −7, giving 10 and −4. Forgetting the negative case gives only 10.
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Question 2
Solve |2x + 1| = 9.
Correct answer: A
From 2x + 1 = 9 we get x = 4; from 2x + 1 = −9 we get 2x = −10, so x = −5. The two solutions are not opposites because of the +1.
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Question 3
How many solutions does |x + 4| = −2 have?
Correct answer: D
An absolute value is a distance, so it is never negative. No value of x makes it equal −2.
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Question 4
Solve 3|x − 2| + 1 = 13.
Correct answer: B
Isolate first: 3|x − 2| = 12, so |x − 2| = 4. Then x − 2 = 4 or x − 2 = −4, giving 6 and −2. Skipping the division by 3 leads to 14 and −10.
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Question 5
Which inequality is equivalent to |x| < 5?
Correct answer: A
Numbers less than 5 units from 0 lie between −5 and 5. A "less than" absolute value inequality always becomes an "and" (between) statement.
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Question 6
Solve |x − 1| ≥ 4.
Correct answer: C
Write x − 1 ≤ −4 or x − 1 ≥ 4, so x ≤ −3 or x ≥ 5. A "greater than" absolute value inequality becomes an "or" statement, not a between statement.
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Question 7
Solve |2x − 6| ≤ 10.
Correct answer: D
Write −10 ≤ 2x − 6 ≤ 10, add 6 to get −4 ≤ 2x ≤ 16, then divide by 2: −2 ≤ x ≤ 8. Stopping before dividing −4 by 2 gives the trap −4.
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Question 8
Which value of x is a solution of |x + 2| > 6?
Correct answer: B
For x = 5, |5 + 2| = 7, and 7 > 6. For −7 the value is |−5| = 5, which is not greater than 6.
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Question 9
A thermostat is set to 70°F, and the room temperature t always stays within 3 degrees of that setting. Which inequality describes t?
Correct answer: B
The distance between t and 70 is |t − 70|, and "within 3 degrees" means that distance is at most 3. Using ≥ would describe temperatures far from the setting.
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Question 10
Which equation has exactly the solutions x = 2 and x = 12?
Correct answer: C
The midpoint of 2 and 12 is 7, and each solution is 5 units from it, so |x − 7| = 5. |x − 5| = 7 gives 12 and −2.
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