An absolute value is an expression that represents the distance of a number from 0. It is marked by two vertical bars on either side of the number, variable, or expression. Anything inside the absolute value bars is called an "argument". Absolute value bars don't work like parentheses, so it's crucial to use them correctly.
Steps
Method 1 of 2: Simplify When the Topic is a Number
Step 1. Determine the expression
Simplifying a numeric argument is a simple process: since the absolute value represents the distance between a number and 0, the answer will always be a positive number. Start by doing the operations between the absolute value bars to determine the expression.
For example, you need to simplify the absolute value of the expression -6 + 3. Since the whole expression is inside the bars of the absolute value, do the addition first. Now the problem is to simplify the absolute value of -3
Step 2. Simplify the absolute value
After you have done all the operations inside the absolute value bars, you can simplify the absolute value. Any number you have as an argument, whether positive or negative, represents a distance from 0, so your answer will be that number, which must be positive.
In the above example, the simplified absolute value is 3. This is true, because the distance between 0 and -3 is 3
Step 3. Use the number line
Optionally, you can write down your answer using the number line. This step can help you visualize absolute values and check your work.
In the example above, your number line will look like this
Method 2 of 2: Simplify When Topic Includes a Variable
Step 1. Simplify an argument consisting of only one variable
If the argument is just a variable, equal to a number, then simplifying is very easy. Since the absolute value represents a distance from 0, the variable can be either the positive number it is equal to, or the negative of that number. There is no way to tell, so you need to include both possibilities in your answer.
- For example, you know that the absolute value of a variable x is equal to 3. You cannot tell if x is positive or negative; you are looking for all numbers whose distance from 0 is 3. So the solutions are 3 and -3.
- If this is the kind of topic you need to simplify, stop here. Are you done. If you have an inequality, continue.
Step 2. Identify the inequalities of the absolute value
If you are given an argument with a variable, expressed as an inequality, other steps are required. Interpret the inequality as a request to find all possible values of the variable.
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For example, you have the following inequality.
This can be interpreted as "Find all numbers whose absolute value is less than 7". In other words, it finds all numbers whose distance from 0 is 7, not including 7 itself. Note that inequality is structured as "less than" rather than "less than or equal to". In the latter case, 7 would also be included.
Step 3. Draw the number line
The first thing to do when working with an inequality of an absolute value is to draw the number line. Mark the points corresponding to the numbers you are working on.
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In the example above, your number line will look like this.
The empty circles indicate the numbers excluded from the final result. Remember: if the inequality is expressed as "greater than or equal to" or "less than or equal to", then these numbers must also be included. In that case, the headbands would be colored.
Step 4. Consider the numbers on the left side of the number line
Since you don't know if the variable is positive or negative, you are dealing with two possible ranges of numbers: those on the left side of the number line and those on the right. First, consider the numbers on the left. Make the variable negative and turn the absolute value bars into parentheses. Solve.
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In the above example you should turn the absolute value bars into parentheses to show that (-x) is less than 7. Multiply both sides of the inequality by -1. Note that when you multiply by a negative number, you have to change the signs of the inequality (from "less than" to "greater than", or vice versa). Inequality will become like this.
Now you know that, for the left side of the number line, x is greater than -7. On the number line, it will be represented like this.
Step 5. Consider the numbers on the right side of the number line
Now you can see the second range of numbers, the positive ones. This is even simpler: make the variable positive and turn the absolute value bars into parentheses.
In the example above you should turn the absolute value bars into parentheses to show that (x) is less than 7. Nothing else is needed in this step. On the number line, it will look like this
Step 6. Find the intersection of the two intervals
Having considered both sides, you need to determine where the solutions overlap. Draw both ranges on the same number line to get the final result.
