Problem 20 - Elementary and Intermediate Algebra

Question

Solve each equation. \(|x|=90\)

Step-by-Step Solution

Answer

The solutions are \\(x = 90\\) and \\(x = -90\\).

Step 1: Understand Absolute Value

The absolute value of a number, denoted by \(|x|\), is the distance of a number from zero on a number line, without regard to direction. For the equation \(|x|=90\), it means that the distance of \(x\) from zero is 90.

Step 2: Set Up Two Equations

Absolute value equations \(|x|=c\) can be solved by setting up two separate equations because the value inside the absolute value can be either positive or negative. Therefore, we set \(x = 90\) and \(x = -90\).

Step 3: Solve Each Equation

We solve the two equations derived from the absolute value: 1. Solve for \(x = 90\): This equation is already solved.2. Solve for \(x = -90\): This equation is also already solved.

Key Concepts

Distance from Zero

When we talk about the absolute value of a number, we're essentially talking about its distance from zero on a number line. This concept is fundamental in mathematics because it helps us understand how numbers relate to each other in terms of size and position without considering direction. For example, \(|x| = 90\) means that a number is 90 units away from zero. This distance can be achieved by moving 90 units to the right, landing at 90, or 90 units to the left, landing at -90. Hence, the absolute value ignores the sign of the number, focusing purely on this distance aspect.
To visualize this, imagine standing at the origin, zero, on a number line. Taking 90 steps forward lands you at 90, while taking 90 steps backward places you at -90. Both are the same distance away from zero but in opposite directions. This is why when solving absolute value equations, the solutions represent distances in both positive and negative directions.

Positive and Negative Solutions

An important thing to grasp about absolute value equations is that they yield two solutions: one positive and one negative. This happens because absolute value measures distance but forgets about direction. So, in the equation \(|x|=90\), \ x \ could be 90 or -90, both of which are exactly 90 units away from zero.
Understanding positive and negative solutions is crucial because it reveals that for any non-zero absolute value \(c\), there are always two points on the number line equidistant from zero. Positive solutions represent the movement to the right side of zero, and negative solutions indicate the movement to the left. A common mistake is to think only about the positive side, but both sides are equally important.
When solving equations, ensure to always consider both possibilities. Double-check your work to confirm you have accounted for both the \(+c\) and \(-c\) solutions.

Solving Equations

Solving absolute value equations involves a few straightforward steps. The main idea is to transform the problem into something familiar by eliminating the absolute value. By doing this, you can work with simpler algebraic equations.
For an equation like \( |x| = 90 \, the first step is to rewrite it without absolute values, as \ x = 90 \ and \ x = -90 \). These two equations represent all possible numbers that are 90 units away from zero. Solving these is a matter of recognizing that they are already in their simplest forms, meaning there's no further calculation needed.

• Identify what number is within the absolute value (90 in this case).
• Write two separate equations that correspond to both positive and negative scenarios.
• Solve each of these simple equations. Often, they are already solved by setting them up.

This method works consistently for all absolute value equations and ensures that you do not miss any possible solutions. Always double-check to make sure you've identified both answers, which completes the task.