How do you write an absolute value inequality?

Absolute value inequalities involve the use of absolute value in mathematical expressions and equations. To write an absolute value inequality, follow these steps:

1. Start by identifying the absolute value expression in the problem.
2. Set up the expression as a greater than or less than statement.
3. Consider two scenarios: one when the expression is positive and one when it is negative.
4. Write the two separate inequalities based on these scenarios.
5. Simplify both inequalities to find the solution set.

To understand this process better, let’s consider an example:

Example:
Write the absolute value inequality for the expression |2x – 3| > 5.

Solution:
Step 1:
Identify the absolute value expression: 2x – 3.

Step 2:
Setup the greater than statement: 2x – 3 > 5.

Step 3:
Consider both cases:
When 2x – 3 is positive: 2x – 3 > 5.
When 2x – 3 is negative: -(2x – 3) > 5.

Step 4:
Simplify the inequalities:
For the positive case: 2x – 3 > 5 –> 2x > 8 –> x > 4.
For the negative case: -(2x – 3) > 5 –> -2x + 3 > 5 –> -2x > 2 –> x < -1. Step 5:
Combine the solutions from both cases: x < -1 or x > 4.

Therefore, the solution to the absolute value inequality |2x – 3| > 5 is x < -1 or x > 4.

Related FAQs:

1. What is an absolute value?

An absolute value represents the distance between a number and zero on a number line, always resulting in a positive value.

2. What does an absolute value inequality mean?

An absolute value inequality is a mathematical statement that involves the inequality symbols (<, >) and the absolute value function.

3. How do you solve an absolute value inequality?

To solve an absolute value inequality, you need to consider both positive and negative cases separately and simplify the resulting inequalities.

4. What is the absolute value of a negative number?

The absolute value of a negative number is its positive counterpart. For example, |-3| = 3.

5. Can an absolute value be negative?

No, the absolute value of a number is always positive or zero.

6. What is the difference between an absolute value equation and an absolute value inequality?

An absolute value equation involves finding the specific values of the variable that satisfy the equation, whereas an absolute value inequality involves finding the range of values that satisfy the inequality.

7. What happens if the absolute value inequality doesn’t have a solution?

If an absolute value inequality has no solution, it means that there is no value of the variable that satisfies the inequality.

8. Can an absolute value inequality have multiple solutions?

Yes, an absolute value inequality can have multiple solutions depending on the range of values that satisfy the inequality.

9. Can an absolute value inequality have an infinite number of solutions?

Yes, an absolute value inequality can have an infinite number of solutions if all values within a particular range satisfy the inequality.

10. What is the solution to an absolute value inequality with no inequality symbols?

An absolute value inequality without inequality symbols is not possible. The inequality symbols indicate the comparison between two values.

11. Can you solve absolute value inequalities graphically?

Yes, absolute value inequalities can be solved graphically by representing the solutions on a number line.

12. How can absolute value inequalities be used in real-life situations?

Absolute value inequalities are frequently used in real-life situations such as determining the range of acceptable values for measurements, time, and finances.

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