Question

Which values of x will make the absolute value equation true?

|x – 4| = 7

A.
{–3, 3}
B.
{–3, 11}
C.
{–11, 11}
D.
no solution

Answer 1

Let's check each answer choice
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Checking choice A
Plug in x = -3
|x-4| = 7
|-3-4| = 7 ... replace x with -3
|-7| = 7
7 = 7
We have a true equation so x = -3 works
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Plug in x = 3
|x-4| = 7
|3-4| = 7 ... replace x with 3
|-1| = 7
1 = 7
The last equation is false so x = 3 is not a solution. We can cross choice A off the list.
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Checking choice B
We already checked x = -3 (see above in choice A). All we need to do is check x = 11
|x-4| = 7
|11-4| = 7 .... replace x with 11
|7| = 7
7 = 7
We have a true equation so x = 11 works
Since x = -3 and x = 11 work, this means we have the proper solution set {-3, 11}. So this means choice B is the final answer
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For the sake of completeness, let's check choice C.
Plug in x = -11
|x-4| = 7
|-11-4| = 7 ... replace x with -11
|-15| = 7
15 = 7
The last equation is false so choice C can be ruled out as well.
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We can cross choice D off the list since we found there is a solution set and it is {-3, 11} back in choice B.
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Let's say that we didn't have these answer choices and it was a fill in the blank type of problem. Here's how to find the two solutions
|x-4|=7
x-4=7 or x-4=-7 ...... break up the absolute value
x-4+4=7+4 or x-4+4=-7+4 .... add 4 to each side (for each equation)
x = 11 or x = -3
x = -3 or x = 11
So the solution set is {-3, 11}
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Final Answer: Choice B) {-3, 11}

Answer 2

Absolute value makes what ever is in those two vertical bars (like how x-4 is) positive. So even if what ends up coming out of those two bars is negative, it is changed to positive by the absolute value. That means x - 4 can equal -7 or 7 because after the absolute value is applied, it's will be made positive so it will equal the positive 7 on the right side of the equation. Let's solve:

x - 4 = 7 (add 4 to both sides to get x by itself)
x = 11 (one of the answers)

x - 4 = -7
x = -3 (other answer)

So the answer is { -3, 11 }