Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the solution sets of the given inequalities. $$ \left|\frac{2 x}{7}-5\right| \geq 7 $$

Step-by-Step Solution

Answer

\(x \leq -7\) or \(x \geq 42\).

Step 1: Understanding the inequality

The inequality presented is \(\left|\frac{2x}{7} - 5\right| \geq 7\). This means that the expression inside the absolute value, \(\frac{2x}{7} - 5\), must be greater than or equal to 7, or less than or equal to -7.

Step 2: Set up the first part of inequality

Set up the first inequality case where: \(\frac{2x}{7} - 5 \geq 7\). We need to solve for \(x\) in this inequality.

Step 3: Solve the first inequality

Add 5 to both sides to isolate the term with \(x\):\[\frac{2x}{7} \geq 12\]Then, multiply both sides by 7:\[2x \geq 84\]Finally, divide by 2:\[x \geq 42\]

Step 4: Set up the second part of inequality

Now, set up the second inequality case where: \(\frac{2x}{7} - 5 \leq -7\). We will solve for \(x\) in this inequality as well.

Step 5: Solve the second inequality

Add 5 to both sides:\[\frac{2x}{7} \leq -2\]Multiply both sides by 7:\[2x \leq -14\]Lastly, divide by 2:\[x \leq -7\]

Step 6: Combine the solution sets

The solution sets from both parts are \(x \geq 42\) and \(x \leq -7\). Since the absolute inequality involves combining the solutions outside the middle values, the final solution set is: \(x \leq -7\) or \(x \geq 42\).

Key Concepts

Understanding Absolute Value Inequalities

An absolute value inequality, such as \( \left|\frac{2x}{7} - 5\right| \geq 7 \), represents a range within which the values of the expression can lie. The absolute value symbol \( |...| \) indicates the distance from zero, meaning it is always positive. Therefore, when solving absolute value inequalities, you need to consider two cases: one for the positive scenario and one for the negative scenario.

In this example, the absolute inequality states that the expression \( \frac{2x}{7} - 5 \) must either be greater than or equal to 7, or less than or equal to -7. This bifurcation ensures that all possible numbers satisfying the inequality conditions are included.

• The first case: \( \frac{2x}{7} - 5 \geq 7 \)
• The second case: \( \frac{2x}{7} - 5 \leq -7 \)

Solving these two inequalities separately will provide the boundaries for our solution set.

Building Solution Sets for Inequalities

When solving inequalities, the solution set encompasses all values of \( x \) that make the inequality true. Once both absolute inequalities are solved separately, the results are combined to form the complete solution set.

In our example, the solution sets for each case after solving were:

• \( x \geq 42 \)
• \( x \leq -7 \)

Since the solution involves values outside the bounded interval (i.e., values to the left of -7 and to the right of 42), the solution set is expressed using the union of two intervals: \( (-\infty, -7] \cup [42, \infty) \). This comprehensive set represents all possible solutions to the absolute value inequality.

Steps in Solving Inequality Equations

Solving inequalities involves a series of logical steps to isolate the variable of interest. The process typically involves:

1. **Setting Up the Inequalities**: Recognize that the absolute expression can lead to two different inequalities, addressing both positive and negative conditions.
2. **Isolating the Variable**: Rearrange the inequality to solve for \( x \):

• For the inequality \( \frac{2x}{7} - 5 \geq 7 \), add 5 to both sides to simplify it to \( \frac{2x}{7} \geq 12 \).
• Multiply each term by 7 to remove the fraction: \( 2x \geq 84 \).
• Finally, divide by 2 to find \( x \geq 42 \).

3. **Repeat for the Second Inequality**: Perform similar steps for the second condition where \( \frac{2x}{7} - 5 \leq -7 \).

Using these steps methodically will help reduce errors and ensure an accurate solution.

Understanding Mathematical Expressions

Mathematical expressions in inequalities are composed of variables, constants, and arithmetic operations (addition, subtraction, multiplication, and division).

Expressions like \( \frac{2x}{7} - 5 \) involve a variable \( x \) which we aim to solve for. Here, the key operations involve:

• **Fractions**: The expression has \( x \) divided by 7, indicating proportionality.
• **Subtraction**: The whole term \( \frac{2x}{7} \) is reduced by 5, setting up the condition for the inequality.

Understanding the role of each component is crucial, as it influences how we manipulate these expressions to solve inequalities.

Consistently practicing manipulating these simpler components will help you get comfortable with solving more complex inequalities.