Absolute value equations and inequalities, in context

Grade 11 Math Worksheet - Building Confidence

Circle the best answer.

1. 1.

   Ari solves |x − 4| = 2x − 5 by splitting it into two branches. Which solution set is correct after checking?

   • No solution: the first branch fails the check, so the equation has none
   • x = 1 only: the first branch, x − 4 = 2x − 5, is the one to use
   • x = 1 and x = 3: each branch gives an answer, so both count
   • x = 3 only: the branch that gives x = 1 fails the check
2. 2.

   Bo solves |2x + 1| = x + 4. One answer is negative. Which solution set is correct after checking?

   • x = 3 and x = −5/3: both pass the check
   • x = 3 and x = −5: the second branch gives −3x = 5
   • Just x = 3: a negative answer cannot be right
   • x = 3 and x = 5/3: the second branch gives 3x = −5, so x is positive
3. 3.

   Esme solves |5 − x| = 2x + 1 and checks each answer. Which solution set is correct?

   • x = −6 only: it comes from the second branch, so it is the answer
   • x = 4/3 and x = 6: the second branch is 5 − x = −2x − 1, so x = 6
   • x = 4/3 only: the answer x = −6 fails the check
   • x = 4/3 and x = −6: both branches give an answer
4. 4.

   Jun gets |x − 6| + 5 = 2 from a sensor model. Which description of its solutions is correct?

   • x = 9 or x = 3: the bars equal 3, so x − 6 = 3 or x − 6 = −3
   • x = 13 or x = −1: the bars equal 7, since 2 + 5 = 7
   • Just x = 9: use the positive branch, x − 6 = 3
   • There are no solutions: the bars would have to equal −3
5. 5.

   Kira bakes loaves that must weigh 450 g, within 2%. Which inequality says a loaf of weight w passes?

   • |w − 450| ≥ 9
   • |w − 9| ≤ 450
   • |w − 450| ≤ 2
   • |w − 450| ≤ 9
6. 6.

   A tire needs 60 psi, and any pressure within 5 psi (ends allowed) is fine. Luz writes the interval of good pressures. Which is it?

   • [50, 70]
   • [55, 65]
   • (55, 65)
   • [55, 60]
7. 7.

   Milo marks every point on a number line at most 3 units from −2. Which inequality describes those points?

   • |x − 2| ≤ 3
   • |x + 3| ≤ 2
   • |x + 2| ≤ 3
   • |x + 2| ≥ 3
8. 8.

   Plants must be more than 4 units from the point x = 1 along a bed. Noah lists the allowed positions. Which is correct?

   • −3 < x < 5
   • x > 5
   • x < −3 or x > 5
   • x < −4 or x > 4
9. 9.

   Ren solves |2x − 6| ≥ 4. Which interval notation gives the solution?

   • (−∞, 2] ∪ [10, ∞)
   • [1, 5]
   • (−∞, 1) ∪ (5, ∞)
   • (−∞, 1] ∪ [5, ∞)
10. 10.

    Tala solves |3 − x| ≤ 2. Which interval notation gives the solution?

    • (1, 5)
    • (−∞, 1] ∪ [5, ∞)
    • [1, 5]
    • [−5, −1]