# How Do You Solve Complex Absolute Value Inequalities?

In summary, the two absolute value problems can be solved using the graph of the function and the equation for zero.
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How to solve these two absolute value problems?

1.
##|3x - 5| > |x + 2|##

My attempt:
From what I read in my textbook, the closest properties of absolute value is the one that uses "equal" sign
##|3x - 5| = |x + 2|##
##3x - 5 = x + 2##
##3x -x = 5 + 2##
##2x = 7##
##x = \frac{7}{2}##

##|3x - 5| = |x + 2|##
##3x - 5 = -(x + 2)##
##3x - 5 = -x - 2##
##3x + x = 5 - 2##
##4x = 3##
##x = \frac{3}{4}##

However, this absolute uses ">" sign. So, how do you solve this one?

2.
|x - 3| + |2x - 8| = 5

Note: this is the absolute value properties from my textbook (please see attached file).

#### Attachments

• AbsValProperties.PNG
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How to solve these two absolute value problems?

1.
##|3x - 5| > |x + 2|##

My attempt:
From what I read in my textbook, the closest properties of absolute value is the one that uses "equal" sign
##|3x - 5| = |x + 2|##
##3x - 5 = x + 2##
##3x -x = 5 + 2##
##2x = 7##
##x = \frac{7}{2}##

##|3x - 5| = |x + 2|##
##3x - 5 = -(x + 2)##
##3x - 5 = -x - 2##
##3x + x = 5 - 2##
##4x = 3##
##x = \frac{3}{4}##

However, this absolute uses ">" sign. So, how do you solve this one?

2.
|x - 3| + |2x - 8| = 5

Note: this is the absolute value properties from my textbook (please see attached file).
Start with the simplest case: what is the solution if both ##2x-5## and ##x+2## are both greater or equal to zero? What condition(s) do you get on ##x## then?

this absolute uses ">" sign.
The abs() function is continuous, so as x varies continuously |3x-5|-|x+2| cannot switch between >0 and <0 without passing through =0.
Thus, the solutions you found for the equals case represent the boundaries for the positive and negative ranges. It is just a matter of testing values of x between and beyond those points.

For (2), you have more cases to consider. How many?

Last edited:
SammyS
Following on from this, I would say the simplest approach is to draw a graph of both functions to see graphically where ##|3x -5|## is greater than ##|x + 2|##.

PeroK said:
Following on from this, I would say the simplest approach is to draw a graph of both functions to see graphically where ##|3x -5|## is greater than ##|x + 2|##.
Perhaps it comes to the same thing, but I look at the points where the individual terms become zero.
In general, we have ##\Sigma a_i|x-b_i|>c##. The ai can be signed.
The function is continuous and consists of straight lines between the points ##x=b_i##. It is not hard to plot the values of the function at those points, and to see what happens as x tends to ±∞.

SammyS

## 1. What is an absolute value?

An absolute value is the distance of a number from zero on a number line. It is always a positive number.

## 2. How do you find the absolute value of a number?

To find the absolute value of a number, you can remove any negative sign in front of the number. If the number is already positive, the absolute value will remain the same.

## 3. What is the symbol used for absolute value?

The symbol used for absolute value is two vertical bars surrounding the number, such as |x|.

## 4. Can the absolute value of a number be negative?

No, the absolute value of a number is always positive. If a negative number is inside the absolute value symbol, it will become positive.

## 5. How is absolute value used in math and science?

Absolute value is used in math and science to represent the magnitude or size of a number or measurement. It is also used to find the distance between two points on a number line or in a coordinate plane.

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