# How Do You Solve Absolute Value Inequalities with Double Variables?

• h00zah
In summary, to solve absolute value inequalities with double variables and bars on both variables, you need to divide the problem into four regions/conditions and find the solutions for each condition. When the absolute value is less than a, it must be -a < x < a. When the absolute value is less than or equal to a, the representation for both rules at once is -a <= x <= a. This method may not have been taught in class or included in your textbook.
h00zah
how do you go about solving abs value inequalities with double variables when the abs value bars are on both the variables?

eg; |x| + or - |y| =, >, <, a

I know that when |x| < a it must be -a < x < a, but when |x| "is less than or equal to" a, how do you represent both rules at once?

this is not h.w, we're doing this in class and we haven't been shown how to do this and there are no examples in my text.

h00zah said:
but when |x| "is less than or equal to" a, how do you represent both rules at once?
You don't
You divide the problem into n regions/conditions.
In this problem it is four conditions; x and y , positive and negative.

Then the interval solutions depend on the condition/region.

Check out
http://www.purplemath.com/modules/absineq.htm

h00zah said:
I know that when |x| < a it must be -a < x < a, but when |x| "is less than or equal to" a, how do you represent both rules at once?
Like this: -a <= x <= a
h00zah said:
this is not h.w, we're doing this in class and we haven't been shown how to do this and there are no examples in my text.

To solve absolute value inequalities with double variables when the absolute value bars are on both variables, you need to follow a few steps:

1. Isolate the absolute value terms: In order to solve the inequality, you need to isolate the absolute value terms on one side of the equation. This can be done by moving all other terms to the other side of the equation.

2. Split the inequality into two: Once the absolute value terms are isolated, you need to split the inequality into two separate inequalities. This is because the absolute value of a number can be either positive or negative, so you need to consider both cases separately.

3. Solve each inequality separately: Now, you can solve each inequality separately by removing the absolute value bars and solving for x and y. Remember to change the inequality sign accordingly when removing the absolute value bars.

4. Combine the solutions: Once you have solved both inequalities, you need to combine the solutions to get the final solution for the original inequality. This can be done by taking the union of the solutions from both inequalities.

For example, if the given inequality is |x| + |y| < a, you would first isolate the absolute value terms to get |x| < a - |y|. Then, you would split the inequality into two: x < a - y and -x < a - y. Finally, you would solve each inequality separately and combine the solutions to get the final solution set for the original inequality.

## What is an absolute value inequality?

An absolute value inequality is an inequality that contains the absolute value of a variable. This means that the variable can have a positive or negative value, but the absolute value ensures that the inequality is always true.

## How do you solve an absolute value inequality?

To solve an absolute value inequality, you must first isolate the absolute value expression on one side of the inequality. Then, you must consider the two possible cases: when the absolute value is positive and when it is negative. Finally, you must solve for the variable in each case and combine the solutions to find the final solution set.

## What is the difference between solving an absolute value equation and an absolute value inequality?

The main difference between solving an absolute value equation and an absolute value inequality is the presence of inequality symbols. In an absolute value equation, the goal is to find the value(s) of the variable that make the equation true. In an absolute value inequality, the goal is to find the value(s) of the variable that make the inequality true.

## Can absolute value inequalities have more than one solution?

Yes, absolute value inequalities can have more than one solution. This is because the absolute value function can produce two different values (positive and negative) for a given input. Therefore, when solving an absolute value inequality, there are typically two possible solutions.

## Why are absolute value inequalities important?

Absolute value inequalities are important because they allow us to represent a range of possible values for a variable. This is useful in many real-world applications, such as determining possible temperature or distance values. Absolute value inequalities also provide a way to compare two quantities and determine which one is greater or less than the other.

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