5 Ways to Solve Absolute Value Inequalities

Understanding Absolute Value Inequalities

Absolute value inequalities are a type of inequality that involves the absolute value of an expression. These inequalities can be quite challenging to solve, but with the right strategies, you can master them. In this article, we will explore five ways to solve absolute value inequalities.

What are Absolute Value Inequalities?

Before we dive into the solutions, let’s first understand what absolute value inequalities are. An absolute value inequality is an inequality that involves the absolute value of an expression. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

5 Ways to Solve Absolute Value Inequalities

Here are five ways to solve absolute value inequalities:

1. The Distance Method

The distance method involves thinking of the absolute value as a distance from zero on the number line. This method is useful when the inequality involves a single absolute value expression.

📝 Note: When using the distance method, make sure to consider both the positive and negative values of the expression.

Example: Solve |x - 3| > 2

Using the distance method, we can rewrite the inequality as:

x - 3 > 2 or x - 3 < -2

Solving for x, we get:

x > 5 or x < 1

2. The Isolation Method

The isolation method involves isolating the absolute value expression on one side of the inequality. This method is useful when the inequality involves multiple absolute value expressions.

Example: Solve |x - 2| + |x - 5| > 6

Using the isolation method, we can isolate the absolute value expressions:

|x - 2| > 6 - |x - 5|

Now, we can use the distance method to solve for x:

x - 2 > 6 - (x - 5) or x - 2 < -6 + (x - 5)

Simplifying, we get:

x > 8 or x < 3

3. The Graphical Method

The graphical method involves graphing the related function on a coordinate plane. This method is useful when the inequality involves multiple absolute value expressions.

Example: Solve |x - 2| + |x - 5| > 6

Using the graphical method, we can graph the related function:

y = |x - 2| + |x - 5|

The graph will show the values of x that satisfy the inequality.

4. The Test Point Method

The test point method involves testing a value of x in the original inequality. This method is useful when the inequality involves multiple absolute value expressions.

Example: Solve |x - 2| + |x - 5| > 6

Using the test point method, we can test a value of x, such as x = 4:

|4 - 2| + |4 - 5| > 6

2 + 1 > 6 (false)

Since the statement is false, we know that x = 4 is not a solution.

5. The Algebraic Method

The algebraic method involves using algebraic manipulations to simplify the inequality. This method is useful when the inequality involves multiple absolute value expressions.

Example: Solve |x - 2| + |x - 5| > 6

Using the algebraic method, we can simplify the inequality:

|x - 2| + |x - 5| > 6

(x - 2) + (x - 5) > 6 (since both expressions are positive)

Combine like terms:

2x - 7 > 6

Add 7 to both sides:

2x > 13

Divide both sides by 2:

x > 6.5

📝 Note: When using the algebraic method, make sure to check your work by plugging in a test value.

Now that we have explored the five ways to solve absolute value inequalities, let’s summarize the key points:

• The distance method involves thinking of the absolute value as a distance from zero on the number line.
• The isolation method involves isolating the absolute value expression on one side of the inequality.
• The graphical method involves graphing the related function on a coordinate plane.
• The test point method involves testing a value of x in the original inequality.
• The algebraic method involves using algebraic manipulations to simplify the inequality.

By using these methods, you can solve absolute value inequalities with confidence.