Problem 6 - Algebra for College Students

Question

Solve each inequality and graph the solutions. \(|x|>3\)

Step-by-Step Solution

Answer

Solution: \(x < -3\) or \(x > 3\). Graph: Open circles on -3 and 3, shading left of -3 and right of 3.

Step 1: Understand Absolute Values

Absolute value represents the distance from zero on the number line. The inequality \(|x| > 3\) states that the distance from zero is greater than 3, leading to two potential scenarios: one where the values of \(x\) are greater than 3 and one where they are less than -3.

Step 2: Split the Inequality

The inequality \(|x| > 3\) can be split into two separate inequalities: 1. \(x > 3\)2. \(x < -3\)

Step 3: Solve Each Inequality

Solve each of the derived inequalities:1. For \(x > 3\): the value of \(x\) can be any number greater than 3.2. For \(x < -3\): the value of \(x\) can be any number less than -3.

Step 4: Graph the Solutions

To graph the solution:- Draw a number line.- For \(x > 3\), shade the part of the line to the right of 3, not including 3 itself (open circle at 3).- For \(x < -3\), shade the part of the line to the left of -3, not including -3 itself (open circle at -3).

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line. It's always a nonnegative number. Whether the number is positive or negative, you take off the sign and keep the magnitude. For example, the absolute value of both 5 and -5 is 5.

In inequalities involving absolute values, like \(|x| > 3\), we're interested in finding numbers whose distance from zero exceeds 3. This can result in two situations:

• Numbers greater than 3
• Numbers less than -3

This is because anything less than -3 or more than 3 fulfills the condition of being over 3 units away from zero. Therefore, absolute values are a powerful way to express distances on the number line.

When you encounter an absolute value inequality, it's typically split into two simpler inequalities to solve each separately.

Number Line

The number line is a visual tool that represents all real numbers as points on a straight line. It's like a ruler but extends infinitely in both directions, with zero in the middle.

Using a number line to solve inequalities helps us visually understand where solutions fit. For example, when solving \(|x| > 3\), you can imagine all points greater than 3 and all points less than -3 marked on this line.

This effective method of representation makes it easier to comprehend the solution sets visually. A number line helps highlight the open intervals where solutions exist, reflecting inequalities ranging across multiple intervals.

Graphing Inequalities

Graphing inequalities on a number line visually represents all possible solutions to the inequality. When graphing \[|x| > 3\], we create a straightforward picture of the intervals that satisfy the inequality. Here's a simple way to do it:

• Draw the number line.
• Locate the point 3 and -3.
• Use an open circle on both points to indicate the value itself is not included (since inequality signs are 'greater than', not 'greater than or equal to').
• Shade the line to the right of 3, and the line to the left of -3, to show all numbers greater than 3 and less than -3.

This graphing process clearly demonstrates the solution set of the inequality on the number line, confirming that any number greater than 3 or less than -3 is part of the solution.