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Chapter 1: Problem 64

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

Short Answer

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

Step by step solution

01

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.
02

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 \(-22/3 < x < 4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Solve equation by the method of your choice. $$ \sqrt{3} x^{2}+6 x+7 \sqrt{3}=0 $$

In Exercises 122鈥�133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. You are choosing between two texting plans. Plan A has a monthly fee of 15 dollar with a charge of 0.08 dollar per text. Plan \(B\) has a monthly fee of 3 dollar with a charge of 0.12 dollar per text. How many text messages in a month make plan A the better deal?

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