Problem 64 - Algebra and Trigonometry

Question

In Exercises 59–94, solve each absolute value inequality. $$ |3 x+5|<17 $$

Step-by-Step Solution

Answer

The solution for the inequality \(|3x+5|<17\) is \(-22/3 < x < 4\).

Step 1: Rewrite the Absolute Value Inequality as a Compound Inequality

The inequality \(|3 x+5|<17\) can be rewritten as \(-17 < 3x + 5 < 17\). This is done as the absolute value of a number is less than 17 means the number is between -17 and 17.

Step 2: Solve the Compound Inequality

To solve for x, we need to isolate x in both inequalities. Starting with the left inequality, when we subtract 5 from all parts of the inequality, we have: \( -17 - 5 < 3x + 5 - 5 < 17 - 5 \), which simplifies to \(-22 < 3x < 12\). Then, to isolate x, we divide all parts of the inequality by 3, giving \(-22/3 < x < 12/3\). This simplifies further to \(-22/3 < x < 4\).

Key Concepts

Compound Inequalities

Compound inequalities are expressions that combine two separate inequalities into one statement. When solving absolute value inequalities, it's essential to understand that the inequality can be rewritten as two parts. Take the example \(|3x+5|<17\). This inequality implies that the expression inside the absolute value lies between -17 and 17.
Let's break it down:

• The expression \(3x+5\) must be greater than -17.
• At the same time, \(3x+5\) must be less than 17.

When these are combined, they form the compound inequality \(-17 < 3x + 5 < 17\). This approach is essential for interpreting and solving absolute value inequalities.

Isolation of Variables

Isolating the variable is a critical step when dealing with inequalities. It involves getting the variable alone on one side of the inequality sign. When we start with the compound inequality \(-17 < 3x + 5 < 17\), our goal is to solve for \(x\).
First, subtract 5 from all parts of the inequality to remove the constant next to the variable. This gives us \(-17 - 5 < 3x + 5 - 5 < 17 - 5\), simplifying to \(-22 < 3x < 12\).
The next step is to divide every term by 3 to fully isolate \(x\). Once divided, we have \(-\frac{22}{3} < x < 4\). Now \(x\) is isolated, and we have a clearer view of the possible values for \(x\).

Solving Inequalities

Solving inequalities is about finding the range of values that satisfy the inequality condition. The procedure is similar to solving equations, but it requires careful attention to the inequality signs.
After forming the compound inequality and isolating the variable, as shown with \(-22 < 3x < 12\), it's essential to manipulate the inequality in a way that preserves the relationship between the numbers.
For example, dividing every part of \(-22 < 3x < 12\) by 3 gives us \(-\frac{22}{3} < x < 4\), which shows the solutions \(x\) can take. Each transformation keeps the inequalities true and allows us to understand \(x\)'s possible values.

Inequality Simplification

Simplifying inequalities transforms them into a more accessible form without changing their meaning. This often involves performing operations that maintain the inequalities' integrity.
In the inequality \(-17 < 3x + 5 < 17\), start by clearing out any constants hindering the view of \(x\). Subtracting 5 from all parts provides the simpler form \(-22 < 3x < 12\).
Next, divide by 3 to make the inequality even simpler: \(-\frac{22}{3} < x < 4\).
Each step simplifies the inequality, making it easier to determine the range of values for \(x\). Through simplification, you can more clearly see and understand the conditions needed to satisfy the inequality.