Problem 58 - Algebra 1: Concepts and Skills

Question

What is the solution of \(|x|=18 ?\) (A) 18 \((B)-18\) (C) 18 and \(-18\) (D) none of these

Step-by-Step Solution

Answer

The solution to \(|x|=18\) is (C) 18 and \(-18\).

Step 1: Solve the positive case

Start off by solving the equation in the case where \(x\) can be a positive value. So when \(x = 18\), it's clear to see that \(|18| = 18\), which is true.

Step 2: Solve the negative case

Next, solve the equation in the case where \(x\) can be a negative value. In this case, when \(x = -18\), we see that \(|-18| = 18\), which is also a true statement.

Step 3: Combine the solutions

Since both \(x = 18\) and \(x = -18\) solve the equation \(|x| = 18\), the solution must be both numbers, 18 and -18.

Key Concepts

Positive and Negative Cases

When you're solving an absolute value equation like \(|x| = 18\), always remember that there are two possible scenarios. Why? Because the absolute value of a number is that number's distance from zero on a number line, which is always positive, no matter if the original number is positive or negative. So, you need to consider both:

• The **positive case**: where the value inside the absolute value, **x**, is positive. Here, \(|x| = x\), which means you're solving \(x = 18\).
• The **negative case**: where the value inside the absolute value, **x**, is negative. Here, the absolute value sign makes it positive, so you're solving \(-x = 18\). Solving for **x**, you get \(x = -18\).

This is why you must consider both cases when dealing with absolute value equations. So the two solutions of the equation \(|x| = 18\) are \(x = 18\) and \(x = -18\).

Absolute Value Equation

An absolute value equation is a type of equation that involves the absolute value function. It is usually written like \(|x| = a\), where \(a\) is a non-negative number. The absolute value symbol \(| |\) takes whatever is inside and makes it non-negative.
The concept behind absolute value is straightforward:

• If **x** is positive or zero, then \(|x| = x\).
• If **x** is negative, then \(|x| = -x\) because the negative sign makes the number positive.

In the equation \(|x| = 18\), **18** is the absolute value, which means the original number could have been either 18 or -18 before the absolute value was taken. To solve such equations, always consider both the positive and negative possibilities.

Solving Equations

Solving equations is one of the fundamental tasks in algebra. When dealing with absolute value equations, it requires understanding the dual nature of absolute values: they account for both the positive and negative origins.
Here's the step-by-step approach for solving an equation like \(|x| = 18\):

• **Identify the absolute value equation.** Notice the absolute value symbol encasing the variable.
• **Solve for the positive case.** Assume \(x\) is positive, so simply set \(x = 18\).
• **Solve for the negative case.** Assume \(x\) is negative, so set \(-x = 18\), which simplifies to \(x = -18\).
• **Combine solutions.** Since both scenarios satisfy the original equation, the solution is \(x = 18\) and \(x = -18\).

This process allows you to systematically tackle absolute value equations, ensuring you consider all possible values **x** could be.