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Absolute Value Equations

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides sample problems which develop concepts related to absolute value equations.

Exercise 1

∣ 4 ∣ = ∣ 4 ∣ = size 12{ lline 4 rline ={}} {}

Exercise 2

∣ − 5 ∣ = ∣ − 5 ∣ = size 12{ lline - 5 rline ={}} {}

Exercise 3

∣ 0 ∣ = ∣ 0 ∣ = size 12{ lline 0 rline ={}} {}

Exercise 4

OK, now, I’m thinking of a number. All I will tell you is that the absolute value of my number is 7.

  • a. Rewrite my question as a math equation instead of a word problem.
  • b. What can my number be?

Exercise 5

I’m thinking of a different number. This time, the absolute value of my number is 0.

  • a. Rewrite my question as a math equation instead of a word problem.
  • b. What can my number be?

Exercise 6

I’m thinking of a different number. This time, the absolute value of my number is –4.

  • a. Rewrite my question as a math equation instead of a word problem.
  • b. What can my number be?

Exercise 7

I’m thinking of a different number. This time, the absolute value of my number is less than 7.

  • a. Rewrite my question as a math inequality instead of a word problem.
  • b. Does 8 work?
  • c. Does 6 work?
  • d. Does −8−8 size 12{ - 8} {} work?
  • e. Does −6−6 size 12{ - 6} {} work?
  • f. Write an inequality that describes all possible values for my number.

Exercise 8

I’m thinking of a different number. This time, the absolute value of my number is greater than 4.

  • a. Rewrite my question as a math inequality instead of a word problem.
  • b. Write an inequality that describes all possible values for my number. (Try a few numbers, as we did in #7.)

Exercise 9

I’m thinking of a different number. This time, the absolute value of my number is greater than −4−4 size 12{ - 4} {}.

  • a. Rewrite my question as a math inequality instead of a word problem.
  • b. Write an inequality that describes all possible values for my number.

Stop at this point and check your answers with me before going on to the next questions.

Exercise 10

∣ x + 3 ∣ = 7 ∣ x + 3 ∣ = 7 size 12{ lline x+3 rline =7} {}

  • a. First, forget that it says “ x+3x+3 size 12{x+3} {}” and just think of it as “a number.” The absolute value of this number is 7. So what can this number be?
  • b. Now, remember that “this number” is x+3x+3 size 12{x+3} {}. So write an equation that says that x+3x+3 size 12{x+3} {} can be <your answer(s) in part (a)>.
  • c. Solve the equation(s) to find what x x can be.
  • d. Plug your answer(s) back into the original equation ∣x+3∣=7∣x+3∣=7 size 12{ lline x+3 rline =7} {} and see if they work.

Exercise 11

4 ∣ 3x − 2 ∣ + 5 = 17 4 ∣ 3x − 2 ∣ + 5 = 17 size 12{4 lline 3x - 2 rline +5="17"} {}

  • a. This time, because the absolute value is not alone, we’re going to start with some algebra. Leave ∣3x−2∣∣3x−2∣ size 12{ lline 3x - 2 rline } {} alone, but get rid of everything around it, so you end up with ∣3x−2∣∣3x−2∣ size 12{ lline 3x - 2 rline } {} alone on the left side, and some other number on the right.
  • b. Now, remember that “some number” is 3x−23x−2 size 12{3x - 2} {}. So write an equation that says that 3x−23x−2 size 12{3x - 2} {} can be <your answer(s) in part a>.
  • c. Solve the equation(s) to find what xx size 12{x} {} can be.
  • d. Plug your answer(s) back into the original equation 4∣3x−2∣+5=174∣3x−2∣+5=17 size 12{4 lline 3x - 2 rline +5="17"} {} and see if they work.

Exercise 12

∣ 3x − 3 ∣ + 5 = 4 ∣ 3x − 3 ∣ + 5 = 4 size 12{ lline 3x - 3 rline +5=4} {}

  • a. Solve, by analogy to the way you solved the last two problems.
  • b. Plug your answer(s) back into the original equation ∣3x−3∣+5=4∣3x−3∣+5=4 size 12{ lline 3x - 3 rline +5=4} {} and see if they work.

Exercise 13

∣x−2∣=2x−10∣x−2∣=2x−10 size 12{ lline x - 2 rline =2x - "10"} {}.

  • a. Solve, by analogy to the way you solved the last two problems.
  • b. Plug your answer(s) back into the original equation ∣x−2∣=2x−10∣x−2∣=2x−10 size 12{ lline x - 2 rline =2x - "10"} {} and see if they work.

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