Problem 80 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

x = 7 or x = -7

Step 1: Isolate the Absolute Value

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

Step 2: 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:\[x = -7\]

Step 3: List All Solutions

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

Key Concepts

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.