Problem 138 - Algebra and Trigonometry

Question

Explaining the Concepts. Describe how to solve an absolute value inequality involving the symbol \(<\). Give an example.

Step-by-Step Solution

Answer

An absolute value inequality \(|x| < a\) is solved by creating two separate inequalities: \(x < a\) and \(x > -a\). The solution for \(x\) is the range between \(a\) and \(-a\) (excluding \(-a\) and \(a\)). For example, for the inequality \(|x-3| < 7\), the solution will be \(-4 < x < 10\).

Step 1: Understanding the absolute value

An absolute value of a number is its distance from zero on the number line. Therefore, it is always positive. For instance, \(|5|= 5\) and \(|-7| = 7\). Any number inside the absolute value bars can be either positive or negative.

Step 2: Formulate the equation

To solve an absolute value inequality such as \(|x| < a\), where \(a\) is a positive number, create two separate inequalities: One is \(x < a\) and the second one is \(x > -a\).

Step 3: Solve the inequalities

Take each inequality separately, solve them by isolating the \(x\). Since the inequalities are simple, the solution is direct.

Step 4: Illustrate with an example

Let's consider an inequality: \(|x-3| < 7\). Rewrite this into two separate inequalities: \(x-3 < 7\) and \(-(x-3) < 7\). Solve each inequality separately, and you will get two solutions: \(x < 10\) and \(x > -4\). Together, the solution set is \(-4 < x < 10\), which means \(x\) can be any number between -4 and 10, not including -4 and 10.

Key Concepts

Solving Inequalities

When tackling inequalities, the process is similar to solving regular equations, but it comes with unique rules you need to follow. An inequality tells us that one side is not equal to the other, but rather less than, greater than, or equal to some extent. There are special symbols involved that guide us in this:

- The less than symbol, \(<\)- The greater than symbol, \(>\)- The less than or equal to symbol, \(\leq\)- The greater than or equal to symbol, \(\geq\)
To start solving, aim to isolate the variable, usually located on one side of the inequality. Identifying this is your first step. But there's a twist: if you multiply or divide by a negative number, remember to flip the inequality symbol. This rule is crucial when handling negative coefficients.

Absolute Value Concept

The term "absolute value" can seem complex but it's quite simple. It measures the distance a number is from zero on a number line, regardless of direction. So, whether it's 5 or -5, both have an absolute value of 5. This number always turns positive.

When given an absolute value inequality such as \(|x-3| < 7\), you're essentially looking at distances. The inequality mandates that the expression inside the absolute bars will remain less than a positive value of 7 from zero. This will help in understanding that there are two scenarios, one where the expression is just under 7 and another where it's just above -7. Understanding being two-sided is key.

Inequality Symbols

Inequality symbols are the heart of comparing magnitudes and they grant us the insight into how numbers or expressions relate to each other in value. These symbols, though simple, carry significant meaning and affect the direction of a problem's solution.

In cases of absolute value inequalities, such as \(|x|

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