Absolute value equations and inequalities, in context

Grade 11 Math Worksheet

Circle the best answer.

1. 1.

   Ari graphs y = |x² − 4x|: the part of the parabola below the x-axis flips up. How many solutions does |x² − 4x| = 3 have?

   • Three: the line y = 3 touches the flipped dip at its top
   • Four: x = 1, 3 and 2 ± √7, as y = 3 also crosses the flipped dip
   • Two: x = 2 ± √7, from x² − 4x = 3 alone
   • Two: x = ±(2 + √7), mirrored across the y-axis
2. 2.

   For f(x) = x² − 6x + 5 = (x − 1)(x − 5), Bo sketches y = f(|x|). Which description of its x-intercepts and shape is right?

   • Two intercepts, x = 1 and 5: the bars just flip negative outputs up
   • Four intercepts, ±1 and ±5: the right half is mirrored across the y-axis
   • Four intercepts, ±1 and ±5, and the graph stays above the x-axis
   • Two intercepts, x = −1 and −5, the mirror images of 1 and 5
3. 3.

   Esme needs road marks x whose distance from mile 1 is less than twice their distance from mile 7: |x − 1| < 2|x − 7|. Which x work?

   • x < 5 or x > 13
   • x < 5, on the side nearer mile 1
   • 5 < x < 13
   • x < 4, the side nearer mile 1 than mile 7
4. 4.

   Jun rewrites g(x) = |x − 1| + |x − 4| without bars, as a piecewise function. Which version is right?

   • 5 − 2x for x < 1, and 2x − 5 for x ≥ 1
   • 2x − 5 for x < 1, 3 for 1 ≤ x ≤ 4, and 5 − 2x for x > 4
   • 5 − 2x for x < 1, −3 for 1 ≤ x ≤ 4, and 2x − 5 for x > 4
   • 5 − 2x for x < 1, 3 for 1 ≤ x ≤ 4, and 2x − 5 for x > 4
5. 5.

   A thermostat set to s keeps the room within 1.5° of s. The room must stay from 18° to 24°. Which settings s does Milo allow?

   • 16.5 ≤ s ≤ 25.5, or |s − 21| ≤ 4.5
   • 18 ≤ s ≤ 24
   • 18.75 ≤ s ≤ 23.25
   • 19.5 ≤ s ≤ 22.5, or |s − 21| ≤ 1.5
6. 6.

   Depots sit at miles 3 and 11. Noah serves spots x nearer depot 3 than depot 11, |x − 3| < |x − 11|, and within 5 miles of it. Which x qualify?

   • −2 ≤ x < 7
   • x < 7
   • 7 < x ≤ 8
   • −2 ≤ x ≤ 7
7. 7.

   Pia graphs y = |x² − 4| and slides a flat line y = k up and down. For which k does |x² − 4| = k have exactly four solutions?

   • k > 0
   • 0 < k ≤ 4
   • 0 < k < 4
   • k > 4
8. 8.

   A sensor reads every spot within 2 m of its position h: |x − h| ≤ 2. Ren needs it to read both x = 5 and x = 8. Which positions h work?

   • 3 ≤ h ≤ 7
   • 6 ≤ h ≤ 7
   • 3 ≤ h ≤ 10
   • 6 < h < 7
9. 9.

   Tala solves |x² − 4| = 3x by splitting it into two branches and checking each answer. Which solution set is right?

   • x = 4 only: the second branch has no real roots
   • x = −1 and x = 4
   • x = −4, −1, 1 and 4
   • x = 1 and x = 4 only

10. Read this first

    Line 1

    x − 1 = 2x + 3, so x = −4. Line 2: x − 1 = −2x + 3, so x = 4/3. Line 3: the solutions are −4 and 4/3.

11. 10.

    Zee checks a worked solution of |x − 1| = |2x + 3|. Which correction is right?

    • Line 3: drop 4/3, since it fails the check, and keep −4
    • Line 2: −(2x + 3) is −2x − 3, which gives x = −2/3
    • No line is wrong: both answers check in the equation
    • Line 2: it should be −x − 1 = 2x + 3, giving x = −4/3