[fs-toc-h2]Intro

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Absolute value refers to how far a number is from zero on the number line, meaning it will always be positive.

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The absolute value of 2 is 2 since its distance from zero is 2.

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The absolute value of -2 is also 2 since its distance from zero is 2.

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Absolute value is denoted by two vertical bars surrounding the expression or number. The absolute value of \(-12\) would be denoted as \(|-12| = 12\).

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All that's important to remember is that anything inside the bars will become positive.

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[fs-toc-h2]Solving equations

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The easiest way to solve for a variable inside absolute value bars is to set that equation equal to both the positive and negative counterparts of the answer. For example, take the equation \(|x+3| = 7\).

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To find both solutions to this equation, set \(x+3\) equal to 7 and \(-7\):

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\[x+3 = 7\]

\[x = 4\]

\[ \text{and} \]

\[x + 3 = -7\]

\[x = -10\]

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So the two solutions to \(|x+3| = 7\) are 4 and -10.

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[fs-toc-h4] Example 1

Whenever given an absolute value equation, always move everything outside of the absolute value bars to the other side of the equal sign.

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\[|n-1| + 1 = 0\]

\[|n-1| = -1\]

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Now we have one side that is all contained within an absolute value and one side that is a negative number. The issue is that no matter what is within the bars, the answer to any absolute value equation MUST be positive. Here it's \(-1\), meaning that there is no possible value for \(n\), making answer choice D the correct answer.

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[fs-toc-h4] Example 2

This question is quite similar to the last one. If we were to set answer choices B, C, and D equal to zero and subtract the one on the outside, you’d have an absolute value equaling a negative number, which gives us no solution.

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This means that the only one that could have some value of \(x\) make the equation equal zero is answer choice A, as you end up with \(|x-1| = 1\), a positive number.

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For more on absolute value, watch this.

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