SOLVING ABSOLUTE VALUE INEQUALITIES
About "Solving absolute value inequalities"
"Solving absolute value inequalities" is the stuff which is being studied by the students who study high school math.
Here we are going to see, "How to solve an absolute value inequality"
We can completely understand "Solving absolute value inequality" by going through the two methods which are explained below.
Method 1 : (Less that or equal to)
Solve the absolute value inequality given below
|x + 2| ≤ 3
Solution :
Let us see how the above mentioned absolute value inequality can be solved.
The picture given below clearly explains you "How the above inequality can be solved"
Let us graph the solution of the first branch x ≤ 1
Let us graph the solution of the second branch x ≥ -5
If we combine the above two graphs, we will get a graph as given below.
From the above graph, the solution for |x + 2| ≤ 3 is
-5 ≤ x ≤ 1
Method 2 : (Greater than or equal to)
Solve the absolute value inequality given below
|x - 3| ≥ 1
Solution :
Let us see how the above mentioned absolute value inequality can be solved.
The picture given below clearly explains you "How the above inequality can be solved"
Let us graph the solution of the first branch x ≥ 4
Let us graph the solution of the second branch x ≤ 2
If we combine the above two graphs, we will get a graph as given below.
From the above graph, the solution for |x - 3| ≥ 1 is
(-∞, 2] U [3, +∞)
Some more problems
Let us look at some practice problems on "Solving absolute value inequalities"
Problem 1 :
Solve the absolute value inequality given below
|2x + 1| ≤ 5
Solution :
In this problem, we have "less than or equal to " symbol.
So, we have to apply "Method 1"
The two branches are
2x + 1 ≤ 5 (or) 2x + 1 ≥ -5
2x ≤ 4 (or) 2x ≥ -6
x ≤ 2 (or) x ≥ -3
Hence the solution is
-3 ≤ x ≤ 2
Let us look at the next problem on "Solving absolute value inequalities"
Problem 2 :
Solve the absolute value inequality given below
|3x + 5| ≥ 7
Solution :
In this problem, we have "greater than or equal to " symbol.
So, we have to apply "Method 2"
The two branches are
3x + 5 ≥ 7 (or) 3x + 5 ≤ -7
3x ≥ 2 (or) 3x ≤ -12
x ≥ 2/3 (or) x ≤ -4
Hence the solution is
(-∞, -4] U [2/3, +∞)
Let us look at the next problem on "Solving absolute value inequalities"
Problem 3 :
Solve the absolute value inequality given below
|x - 1| +2 ≤ 5
Solution :
First Let us write the given absolute value inequality in standard form.
|x - 1| + 2 ≤ 5 -----------------> |x - 1| ≤ 3
In this problem, we have "less than or equal to" symbol.
So, we have to apply "Method 1"
The two branches are
x - 1 ≤ 3 (or) x - 1 ≥ -3
x ≤ 4 (or) x ≥ -2
Hence the solution is
-2 ≤ x ≤ 4
Let us look at the next problem on "Solving absolute value inequalities"
Problem 4 :
Solve the absolute value inequality given below
|2x - 3| - 5 ≥ 7
Solution :
First Let us write the given absolute value inequality in standard form.
|2x - 3| - 5 ≥ 7 -----------------> |2x - 3| ≥ 12
In this problem, we have "greater than or equal to " symbol.
So, we have to apply "Method 2"
The two branches are
2x - 3 ≥ 12 (or) 2x - 3 ≤ -12
2x ≥ 15 (or) 2x ≤ -9
x ≥ 15/2 (or) x ≤ -9/2
Hence the solution is
(-∞, -9/2] U [15/2, +∞)
Let us look at the next problem on "Solving absolute value inequalities"
Problem 5 :
Solve the absolute value inequality given below
2|x + 1| ≤ 6
Solution :
First Let us write the given absolute value inequality in standard form.
2|x + 1| ≤ 6 -----------------> |x + 1| ≤ 3
In this problem, we have "less than or equal to" symbol.
So, we have to apply "Method 1"
The two branches are
x + 1 ≤ 3 (or) x + 1 ≥ -3
x ≤ 2 (or) x ≥ -4
Hence the solution is
-4 ≤ x ≤ 2
Let us look at the next problem on "Solving absolute value inequalities"
Problem 6 :
Solve the absolute value inequality given below
5|x - 3| ≥ 15
Solution :
First Let us write the given absolute value inequality in standard form.
5|x - 3| ≥ 15 -----------------> |x - 3| ≥ 5
In this problem, we have "greater than or equal to " symbol.
So, we have to apply "Method 2"
The two branches are
x - 3 ≥ 5 (or) x - 3 ≤ -5
x ≥ 8 (or) x ≤ -2
Hence the solution is
(-∞, -2] U [8, +∞)
Let us look at the next problem on "Solving absolute value inequalities"
Problem 7 :
Solve the absolute value inequality given below
2|x + 3| + 5 ≤ 13
Solution :
First Let us write the given absolute value inequality in standard form.
2|x + 3|+5 ≤ 13 --------->2|x + 3| ≤ 8 ------->|x+3| ≤ 4
In this problem, we have "less than or equal to" symbol.
So, we have to apply "Method 1"
The two branches are
x + 3 ≤ 4 (or) x + 3 ≥ -4
x ≤ 1 (or) x ≥ -7
Hence the solution is
-7 ≤ x ≤ 1
Let us look at the next problem on "Solving absolute value inequalities"
Problem 8 :
Solve the absolute value inequality given below
5|x +7| - 2 ≥ 18
Solution :
First Let us write the given absolute value inequality in standard form.
5|x+7|-2 ≥ 18 ---------> 5|x+7| ≥ 20 --------->|x+7|≥4
In this problem, we have "greater than or equal to " symbol.
So, we have to apply "Method 2"
The two branches are
x + 7 ≥ 4 (or) x + 7 ≤ -4
x ≥ -3 (or) x ≤ -11
Hence the solution is
(-∞, -11] U [-3, +∞)
Let us look at the next problem on "Solving absolute value inequalities"
Problem 9 :
Solve the absolute value inequality given below
|x + 3| < 13
Solution :
In this problem, we have "less than or equal to" symbol.
So, we have to apply "Method 1"
The two branches are
x + 3 < 13 (or) x + 3 > -13
x < 10 (or) x > -16
Hence the solution is
-16 < x < 10
Let us look at the next problem on "Solving absolute value inequalities"
Problem 10 :
Solve the absolute value inequality given below
|x +7| > 18
Solution :
In this problem, we have "greater than or equal to " symbol.
So, we have to apply "Method 2"
The two branches are
x + 7 > 18 (or) x + 7 < -18
x > 11 (or) x < -25
Hence the solution is
(-∞, -25) U (11, +∞)
Apart from the stuff and problems on "Solving absolute value inequalities, You can also visit the following pages.
Solving absolute value Equations
Graphing absolute value equations
Solving absolute value equations worksheet
Solving equations with absolute values on both the sides of the equal sign
If you want to know more about "Solving absolute inequalities", please click here.
