Case 1 :
Inequality in the form |x - a| < r.
We can write the given absolute value inequality into two branches as shown below.
x - a < r x < a + r |
x - a > - r x > a - r |
Combine the above two inequalities.
(a - r) < x < (a + r)
So, the solution to |x - a| < r is
(a-r, a+r)
Case 2 :
Inequality in the form |x - a| > r.
We can write the given absolute value inequality into two branches as shown below.
x - a > r x > a + r |
x - a < - r x < a - r |
We can cannot combine the above two inequalities.
So, the solution to |x - a| > r is
(-∞, a - r) U (a + r, ∞)
Case 3 :
Inequality in the form |x - a| ≤ r.
We can write the given absolute value inequality into two branches as shown below.
x - a ≤ r x ≤ a + r |
x - a ≥ - r x ≥ a - r |
Combine the above two inequalities.
(a - r) ≤ x ≤ (a + r)
So, the solution to |x - a| ≤ r is
[a-r, a+r]
Case 4 :
Inequality in the form |x - a| ≥ r.
We can write the given absolute value inequality into two branches as shown below.
x - a ≥ r x ≥ a + r |
x - a ≤ - r x ≤ a - r |
We can cannot combine the above two inequalities.
So, the solution to |x - a| ≥ r is
(-∞, a - r] U [a + r, ∞)
Case 5 :
Inequality in the form |x - a| > - r.
Here, the solution is all real numbers.
Because, the absolute value of any number will be positive and also it is greater than a negative value.
Case 6 :
Inequality in the form :
|x - a| < - r
or
|x - a| ≤ - r
Here, there is no solution.
Because, the absolute value of any number will be positive and it can never be less than or equal to a negative value.
Question 1 :
Solve for x :
|3 - x| < 7
Solution :
|3 - x| < 7
We can write the above absolute value inequality into two branches as shown below.
3 - x < 7 - x < 4 x > -4 |
3 - x > -7 -x > -10 x < 10 |
Combine the above two inequalities.
-4 < x < 10
So, the solution is
(-4, 10)
Question 2 :
Solve for x :
|4x - 5| > -2
Solution :
|4x - 5| > -2
Here, the solution is all real numbers.
Because absolute value of any number will be positive and also it is greater than a negative value.
Question 3 :
Solve for x :
|3 - (3x/4)| ≤ 1/4
Solution :
|3 - (3x/4)| ≤ 1/4
We can write the above absolute value inequality into two branches as shown below.
3 - (3x/4) ≤ 1/4 -3x/4 ≤ -11/4 3x/4 ≥ 11/4 3x ≥ 11 x ≥ 11/3 |
3 - (3x/4) ≥ -1/4 -3x/4 ≥ -13/4 3x/4 ≤ 13/4 3x ≤ 13 x ≤ 13/3 |
Combine the above two inequalities.
11/3 ≤ x ≤ 13/3
So, the solution is
[11/3, 13/3]
Question 4 :
Solve for x :
|x| - 10 < -3
Solution :
|x| - 10 < -3
Add 10 to each side.
|x| < 7
We can write the above absolute value inequality into two branches as shown below.
x > 7 |
x < -7 |
Combine the above two inequalities.
-7 < x < 7
So, the solution is
(-7, 7)
Question 5 :
Solve (1/|2x - 1|) < 6 and express the solution using interval notation.
Solution :
(1/|2x - 1|) < 6
Multiply each side by |2x - 1|.
1 < 6|2x - 1|
Divide each side by 6.
1/6 < |2x - 1|
|2x - 1| > 1/6
We can write the above absolute value inequality into two branches as shown below.
2x - 1 > 1/6 12x - 6 > 1 12x > 7 x > 7/12 |
2x - 1 < -1/6 12x - 6 < -1 12x < 5 x < 5/12 |
We can not combine the above two inequalities.
So, the solution is
(-∞, 5/12) U (7/12, ∞)
Question 6 :
Solve −3|x| + 5 ≤ −2 and graph the solution set in a number line
Solution :
-3|x| + 5 ≤ -2
Subtract 5 from each side.
-3|x| ≤ -7
Divide each side by (-3).
|x| ≥ 7/3
We can write the above absolute value inequality into two branches as shown below.
x ≥ 7/3 |
x ≤ -7/3 |
We can not combine the above two inequalities.
So, the solution is
(-∞, -7/3] U [7/3, ∞)
Question 7 :
Solve 2|x + 1| - 6 ≤ 7 and graph the solution set in a number line.
Solution :
2|x + 1| - 6 ≤ 7
Add 6 to each side.
2|x + 1| ≤ 13
Divide each side by 2.
|x + 1| ≤ 13/2
x + 1 ≤ 13/2 x ≤ 11/2 |
x + 1 ≥ -13/2 x ≥ -15/2 |
Combine the above two inequalities.
-15/2 ≤ x ≤ 11/2
So, the solution is
[-15/2, 11/2]
Question 8 :
Solve (1/5) |10x − 2| < 1.
Solution :
(1/5) |10x − 2| < 1
Multiply each side by 5.
|10x - 2| < 5
10x - 2 < 5 10x < 7 x < 7/10 |
10x - 2 > -5 10x > -3 x > -3/10 |
Combine the above two inequalities.
-3/10 < x < 7/10
So, the solution is
(-3/10, 7/10)
Question 9 :
Solve for x :
|5x - 12| < -2
Solution :
Here, there is no solution.
Because, absolute value of any number will be positive and it can never be less than or equal to a negative value.
Question 10 :
Solve for x :
|x| < 0
Solution :
Here, there is no solution.
Because, absolute value of any number will be positive and it can never be a negative value.
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