Problem 56 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality. Then graph the solution set on the real number line. \(|5 x|>10\)

Step-by-Step Solution

Answer

The solution to the inequality \(|5x| > 10\) is \(x > 2\) or \(x < -2\)

Step 1: Divide the inequality into two parts

We start by separating the inequality into two parts based on the properties of absolute values. This gives us \(5x > 10\) and \(5x < -10\)

Step 2: Solve the first inequality

To solve the first inequality \(5x > 10\), we divide both sides by 5, yielding \(x > 2\)

Step 3: Solve the second inequality

For the second inequality \(5x < -10\), we also divide both sides by 5, resulting in \(x < -2\)

Step 4: Graph the solution set in the real number line

The solution to the inequality \(|5x| > 10\) is \(x > 2\) or \(x < -2\). This can be represented on a real number line with two rays pointing outwards from -2 and 2

Key Concepts

Understanding Absolute Value

Absolute value, denoted by vertical bars around a number or expression (like \(|x|\)), represents the distance of that number from zero on the real number line. This means it is always a non-negative number. When dealing with absolute values in inequalities, like \(|5x| > 10\), you consider two scenarios: the expression inside the absolute value is either greater than 10 or less than -10.

• If the expression inside is positive, you solve normally for \(5x > 10\).
• If the expression inside is negative, you solve for \(5x < -10\).

Understanding these two outcomes helps to break down the inequality into more manageable parts.

Plotting on the Real Number Line

The real number line is a visual tool that helps represent numbers as points on a line. It is essential when you want to graph solutions to inequalities. When you finish solving each part of the absolute value inequality, you plot the solutions on this line.

For our example, \(x > 2\) and \(x < -2\) are the solutions. This means you plot two rays:

• One ray starts at \(2\) and goes to positive infinity.
• The other starts at \(-2\) and goes to negative infinity.

You use open circles at \(2\) and \(-2\) to indicate these values are not included in the solution set.

How to Graph Inequalities

Graphing inequalities involves showing the solution of an inequality on the real number line or a coordinate plane. For inequalities like \(|5x| > 10\), you are finding where the expression is not equal to a single point, but a range of values.

You shade regions on the number line to represent this range. In this case:

• Shade to the right of \(2\) for \(x > 2\).
• Shade to the left of \(-2\) for \(x < -2\).

These shaded areas show where the inequality holds true and help easily visualize the solution set.

Solving Algebraic Inequalities

Algebraic solutions to inequalities involve isolating the variable just like solving equations. However, with inequalities, you must remember that multiplying or dividing by a negative number reverses the inequality sign.

Here’s how you handle the inequality \( |5x| > 10 \):

• Split it into two inequalities due to the absolute value: \(5x > 10\) and \(5x < -10\).
• For \(5x > 10\), divide by 5 to get \(x > 2\).
• For \(5x < -10\), dividing by 5 results in \(x < -2\).

This approach ensures you find all possible solutions that satisfy the original absolute value inequality.