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Chapter 1: Problem 80

Solve each equation or inequality. $$ |x|+3=10 $$

Short Answer

Expert verified
x = 7 or x = -7

Step by step solution

01

Isolate the Absolute Value

First, isolate the absolute value expression by subtracting 3 from both sides of the equation:\[|x| + 3 - 3 = 10 - 3\]\[|x| = 7\]
02

Create Two Separate Equations

The absolute value of a number is equal to 7, so set up two separate equations to solve for x. One for the positive case and one for the negative case:For the positive case:\[x = 7\]For the negative ca se:\[x = -7\]
03

List All Solutions

Combine the solutions from both the positive and negative case:\[x = 7\] or \[x = -7\]

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Key Concepts

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

absolute value
The concept of absolute value is essential in understanding how to solve absolute value equations. Absolute value refers to the distance a number is from zero on the number line, without considering direction.
For example, the absolute value of both -7 and 7 is 7 because they are both 7 units away from zero.
Absolute value is denoted by vertical bars, for instance, \(| x | \).
This means that \(| x | = 7\) implies two cases: where x is +7 and -7.
equation solving
Solving equations involves finding values for variables that make the equation true. When it comes to absolute value equations, the process involves a few clear steps.
Start by isolating the absolute value on one side of the equation. In our exercise, this involves subtracting 3 from both sides to get \(|x| = 7\).
Once isolated, you realize \(|x| = 7\) means x can be either 7 or -7. Hence, you create two separate equations: \[ x = 7 \] and \[ x = -7 \].
Finally, list all solutions: \[ x = 7 \] or \[ x = -7 \].
isolation of variables
Isolation of variables is a fundamental technique in solving equations. It means getting the variable of interest by itself on one side of the equation.
Let's see this in action for the equation \(| x | + 3 = 10\). To isolate \| x |\, subtract 3 from both sides:
\[ | x | + 3 - 3 = 10 - 3 \] simplifies to \[ | x | = 7 \].
Now that \(| x |\) is isolated, we can proceed to solve for x by creating two cases: \[ x = 7 \] and \[ x = -7 \].
This isolation step is crucial because it simplifies the equation, making it easier to identify all solutions.

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