Problem 19 - Elementary and Intermediate Algebra

Question

Solve each equation. \(|x|=23\)

Step-by-Step Solution

Answer

The solutions are \(x = 23\) and \(x = -23\).

Step 1: Understand the Absolute Value

The absolute value of a number represents its distance from zero on the number line. For \(|x| = 23\), this means \(|x|\) is 23 units away from 0.

Step 2: Consider Positive and Negative Cases

Since the absolute value of \(x\) is 23, \(x\) can either be 23 or -23. This gives us two potential solutions for the equation \(|x| = 23\): \(x = 23\) and \(x = -23\).

Step 3: Verify the Solutions

Substitute \(x = 23\) and \(x = -23\) back into the original equation \(|x| = 23\) to ensure both satisfy the equation. Since \(|23| = 23\) and \(|-23| = 23\), both are correct solutions.

Key Concepts

Solving Equations

Solving equations is the process of finding values that satisfy the given equation. In the case of absolute value equations like \(|x| = 23\), we look for values that make the statements true. \(|x|\) represents the absolute value, which tells us how far a number is from zero, without considering direction. We want to find all possible values of \(x\) that maintain the absolute distance of 23 units from zero. This involves considering both the positive and negative scenarios for \(x\), as any value that is 23 units away from zero makes the equation true.

Distance from Zero

The absolute value of a number indicates its distance from zero on a number line. This concept is key when solving equations involving absolute values. For the equation \(|x| = 23\), the absolute value \(|x|\) suggests \(x\) is exactly 23 units away from zero. Understanding this helps us visualize that the value of \(x\) can be positioned 23 units to the right and to the left of zero. In simpler terms, \(x\) can be 23 or -23; both distances result in an absolute value of 23 when considering its placement relative to zero.

Positive and Negative Cases

When solving absolute value equations, it's important to address both the positive and negative cases. This means that when we have \(|x| = 23\), we consider two possibilities for \(x\):

• \(+x = 23\)
• \(-x = 23\)

Both scenarios need exploration because absolute value doesn't care whether the number is positive or negative. It simply measures how far away a number is from zero. Therefore, we solve for both \(x = 23\) and \(x = -23\) to cover all parts of the equation's requirements.

Verification of Solutions

Verifying solutions for equations is crucial to ensure they are correct. After solving the equation \(|x| = 23\) and proposing that \(x\) can be either 23 or -23, we need to check if these solutions are valid. This is done by substituting them back into the original equation:

• For \(x = 23\), sub it back to get \(|23| = 23\)
• For \(x = -23\), sub it back to get \(|-23| = 23\)

Both substitutions lead to true statements, confirming that 23 and -23 indeed satisfy the original equation. Verification of solutions confirms that our process of solving was accurate and thorough.