Problem 79 - Algebra for College Students

Question

Solve each equation or inequality. $$ |x|-1=4 $$

Step-by-Step Solution

Answer

x = 5 or x = -5

Step 1: Set up the Absolute Value Equation

An absolute value equation \(|x| - 1 = 4\) can be solved by isolating the absolute value expression.

Step 2: Isolate the Absolute Value

Add 1 to both sides of the equation to isolate the absolute value expression: \[ |x| = 5 \]

Step 3: Split the Equation

Since the absolute value of \(x\) equals 5, this means \(x\) could be 5 or -5. So, split the equation into two cases: \[ x = 5 \] and \[ x = -5 \]

Step 4: Verify Solutions

Check both solutions by substituting back into the original equation to ensure they satisfy \(|x| - 1 = 4\). \[ |5| - 1 = 5 - 1 = 4 \] and \[ |-5| - 1 = 5 - 1 = 4 \]. Both check out.

Key Concepts

absolute value equations

Absolute value equations involve expressions within absolute value signs. The absolute value of a number is its distance from zero on the number line, regardless of direction. This means the absolute value of both 5 and -5 is 5. When solving an absolute value equation, you need to find all possible values that satisfy the equation.

For instance, consider the equation \(|x|-1=4\). Here, \(|x|\) represents the absolute value. The first step is to isolate the absolute value expression.

isolating absolute value

To isolate the absolute value in the equation \(|x|-1=4\), you need to get \(|x|\) by itself on one side of the equation. This involves basic algebraic manipulation. Here's how you can do it:

• Add 1 to both sides of the equation. This eliminates the -1 on the left side.
• You end up with \[ |x| = 5 \]. Now the absolute value is isolated.

You've simplified the equation to a point where you can now 'split' it to solve for \(|x|\).

splitting equations

Once the absolute value is isolated, you need to consider both positive and negative solutions. Because \(|x|=5\), this implies that \[ x=5 \] or \[ x=-5 \]. This process is known as 'splitting' the equation.

• So, from \[ |x|=5 \], you derive two separate equations: \[ x=5 \] and \[ x=-5 \].
• Each of these equations represents a case where the absolute value condition is satisfied.

Now you have potential solutions that need to be verified.

verifying solutions

The final step in solving an absolute value equation is to verify your solutions. Substituting your solutions back into the original equation ensures they are correct.

Let's verify the solutions \[ x=5 \] and \[ x=-5 \]:

• Substitute \[ x=5 \] into the original equation: \[ |5|-1=5-1=4 \], which checks out.
• Substitute \[ x=-5 \] into the original equation: \[ |-5|-1=5-1=4 \], which also checks out.

Both solutions verify correctly, confirming they are accurate solutions to the equation \(|x|-1=4\).