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Chapter 1: Problem 6

Solve each equation. $$ |x|=14 $$

Short Answer

Expert verified
The solutions are \(x = 14\) and \(x = -14\).

Step by step solution

01

Identify the Absolute Value Equation

The given equation is \(|x| = 14\). An absolute value equation of the form \(|x| = a\) has two solutions because the absolute value of a number is always positive.
02

Set Up the Two Possible Equations

Given \(|x|=14\), the two possible values for \(x\) that satisfy this equation are \(x = 14\) and \(x = -14\), because both 14 and -14 have an absolute value of 14.
03

Solve for x in Each Equation

- For the first equation, it is already solved: \(x = 14\).- For the second equation, it is also already solved: \(x = -14\).
04

Verify the Solutions

Verify each solution to ensure they satisfy the original equation: \(|14| = 14\) and \(|-14| = 14\), which confirms both solutions are correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

absolute value
An absolute value represents the distance of any number from zero on a number line. It is always a non-negative number. The notation for absolute value is vertical bars on either side of the number, like this: \(|x|\).

For example:
- \(|5| = 5\)
- \(|-5| = 5\)

Notice how both 5 and -5 have the same absolute value. This happens because absolute value looks at magnitude, not direction.
equation solving
Solving an equation means finding the value(s) of the variable that make the equation true. With absolute value equations, like \(|x| = 14\), we need to consider both positive and negative solutions. This is because both positive 14 and negative 14 have the same absolute value.

Here's how you can solve absolute value equations:
  • Identify the absolute value equation: \(|x| = 14\)
  • Set up two possible equations: \(x = 14\) and \(x = -14\)
  • Solve each equation individually. In this case, the solutions are simple: \(x = 14\) and \(x = -14\).
positive and negative solutions
Any absolute value equation \(|x| = a\) has two potential solutions: one positive and one negative. This gives us two equations to solve:
  • \(x = a\)
  • \(x = -a\)
For example, in the equation \(|x|=14\):
  • The first solution is \(x = 14\), because the absolute value of 14 is 14.
  • The second solution is \(x = -14\), because the absolute value of -14 is also 14.
Verification Step:
We confirm each solution by substituting back into the original equation to ensure it holds true:
  • \(|14| = 14\), which is true.
  • \(|-14|=14\), which is also true.

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