Absolute Value practice questions

Absolute Value questions

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1. 1Absolute valueFoundation

   If f(x) = |x - 6|, what is the value of f(2)?If f(x) = |x - 6|, what is the value of f(2)?

   • A-4-4
   • B88
   • C44
   • D22
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   Answer: 44

   Substituting gives |2 - 6| = |-4| = 4. Absolute value returns a distance, so it is never negative, which rules out the first choice without any arithmetic.Substituting gives |2 - 6| = |-4| = 4. Absolute value returns a distance, so it is never negative, which rules out the first choice without any arithmetic.

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2. 2Absolute valueStandard

   How many values of x satisfy the equation |x - 4| = 7?How many values of x satisfy the equation |x - 4| = 7?

   • A22
   • BInfinitely manyInfinitely many
   • C00
   • D11
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   Answer: 22

   An absolute value equals 7 when the expression inside is 7 or -7. Solving x - 4 = 7 gives x = 11 and x - 4 = -7 gives x = -3, so there are two solutions.An absolute value equals 7 when the expression inside is 7 or -7. Solving x - 4 = 7 gives x = 11 and x - 4 = -7 gives x = -3, so there are two solutions.

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3. 3Absolute valueStandard

   Which inequality describes all values within 4 units of 9 on the number line?Which inequality describes all values within 4 units of 9 on the number line?

   • A|x - 4| ≤ 9|x - 4| ≤ 9
   • B|9 - 4| ≤ x|9 - 4| ≤ x
   • C|x + 9| ≤ 4|x + 9| ≤ 4
   • D|x - 9| ≤ 4|x - 9| ≤ 4
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   Answer: |x - 9| ≤ 4|x - 9| ≤ 4

   Distance from 9 is written |x - 9|, and within 4 units means that distance is at most 4. The subtraction names the center, not the radius.Distance from 9 is written |x - 9|, and within 4 units means that distance is at most 4. The subtraction names the center, not the radius.

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4. 4Absolute valueStandard

   What is the sum of all solutions to |2x - 5| = 9?What is the sum of all solutions to |2x - 5| = 9?

   • A22
   • B99
   • C-2-2
   • D55
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   Answer: 55

   The expression inside the bars is 9 or -9. From 2x - 5 = 9, x = 7. From 2x - 5 = -9, x = -2. Their sum is 7 + (-2) = 5. Solving only the first equation is what leaves you with a single answer.The expression inside the bars is 9 or -9. From 2x - 5 = 9, x = 7. From 2x - 5 = -9, x = -2. Their sum is 7 + (-2) = 5. Solving only the first equation is what leaves you with a single answer.

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5. 5Absolute valueStandard

   Which inequality describes every number whose distance from 7 on the number line is at most 3?Which inequality describes every number whose distance from 7 on the number line is at most 3?

   • A|x + 7| ≤ 3|x + 7| ≤ 3
   • B|x - 3| ≤ 7|x - 3| ≤ 7
   • C|x - 7| ≤ 3|x - 7| ≤ 3
   • D|x - 7| ≥ 3|x - 7| ≥ 3
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   Answer: |x - 7| ≤ 3|x - 7| ≤ 3

   Distance from 7 is written |x - 7|, and "at most 3" makes it less than or equal to 3. The plus sign would measure distance from -7, and the reversed inequality describes the numbers further than 3 away.Distance from 7 is written |x - 7|, and "at most 3" makes it less than or equal to 3. The plus sign would measure distance from -7, and the reversed inequality describes the numbers further than 3 away.

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6. 6Absolute valueChallenge

   How many values of x satisfy |2x - 6| = -4?How many values of x satisfy |2x - 6| = -4?

   • A11
   • BInfinitely manyInfinitely many
   • C00
   • D22
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   Answer: 00

   An absolute value is never negative, so no value of x can make it equal -4. Solving the two cases anyway produces answers that fail when checked.An absolute value is never negative, so no value of x can make it equal -4. Solving the two cases anyway produces answers that fail when checked.

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7. 7Absolute valueChallenge

   For how many integer values of x is |x - 4| < 3 true?For how many integer values of x is |x - 4| < 3 true?

   • A33
   • B77
   • C66
   • D55
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   Answer: 55

   The inequality says x - 4 lies strictly between -3 and 3, so x lies strictly between 1 and 7. The integers in that range are 2, 3, 4, 5 and 6 — five of them. Counting 1 and 7 as well gives 7, but the inequality is strict.The inequality says x - 4 lies strictly between -3 and 3, so x lies strictly between 1 and 7. The integers in that range are 2, 3, 4, 5 and 6 — five of them. Counting 1 and 7 as well gives 7, but the inequality is strict.

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8. 8Absolute valueChallenge

   What is the sum of all solutions to |3x + 6| = 15?What is the sum of all solutions to |3x + 6| = 15?

   • A33
   • B-7-7
   • C1010
   • D-4-4
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   Answer: -4-4

   The expression inside is 15 or -15. From 3x + 6 = 15 comes x = 3; from 3x + 6 = -15 comes x = -7. Their sum is -4. The first two choices are the individual solutions, which is what stopping after one branch leaves you with.The expression inside is 15 or -15. From 3x + 6 = 15 comes x = 3; from 3x + 6 = -15 comes x = -7. Their sum is -4. The first two choices are the individual solutions, which is what stopping after one branch leaves you with.

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9. 9Absolute valueChallenge

   Which describes every solution to |x + 2| ≥ 6?Which describes every solution to |x + 2| ≥ 6?

   • Ax ≥ 4x ≥ 4
   • Bx ≤ -8 or x ≥ 4x ≤ -8 or x ≥ 4
   • C-8 ≤ x ≤ 4-8 ≤ x ≤ 4
   • Dx ≤ -8x ≤ -8
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   Answer: x ≤ -8 or x ≥ 4x ≤ -8 or x ≥ 4

   The expression inside is at least 6 away from zero in either direction, so x + 2 ≥ 6 or x + 2 ≤ -6, giving x ≥ 4 or x ≤ -8. A "greater than" absolute value splits into two rays; only "less than" gives a single interval.The expression inside is at least 6 away from zero in either direction, so x + 2 ≥ 6 or x + 2 ≤ -6, giving x ≥ 4 or x ≤ -8. A "greater than" absolute value splits into two rays; only "less than" gives a single interval.

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10. 10Absolute valueChallenge

    What is the product of all solutions to |x - 9| = 4?What is the product of all solutions to |x - 9| = 4?

    • A1818
    • B6565
    • C3636
    • D1313
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    Answer: 6565

    The expression inside is 4 or -4, giving x = 13 and x = 5. Their product is 65. The choices 13 and 18 are one solution and the sum, both of which come from stopping early.The expression inside is 4 or -4, giving x = 13 and x = 5. Their product is 65. The choices 13 and 18 are one solution and the sum, both of which come from stopping early.

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