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.
  • #1
askor
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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

I don't understand at all of absolute value problem like above one. Please help me.

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

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  • #2
askor said:
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

I don't understand at all of absolute value problem like above one. Please help me.

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?
 
  • #3
askor said:
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?
 
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  • #4
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|##.
 
  • #5
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 ±∞.
 
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Related to How Do You Solve Complex Absolute Value Inequalities?

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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