How to Solve Inequalities – Quick Review and Practice
 Posted by Brian Stocker
 Date Published March 19, 2019
 Date modified June 16, 2020
 Comments 4 comments
Solving linear Inequalities – A quick Tutorial
Basic linear inequalities have one of the following forms:
ax + b > 0
ax + b < 0
ax + b > 0
ax + b < 0
where a and b are some real numbers. Our solution to any of these inequalities would be some interval. Let’s see one simple example:
2x – 10 > 16
2x > 16 + 10
2x > 26
x > 26/2
x > 13
So, the interval here is: (3, +∞)
If we have a case where –x is lesser or greater than some number, then we multiply the whole inequality by 1, where the sign of inequality also changes:
3x + 9 < 12
3x < 12 – 9
3x < 3
x < 3/(1)
x > 3
So, the interval here is: [3, +∞) Notice the difference in the brackets. This is because this interval contains number 3.
Let’s see a little more complex example:
x / (x + 1) > 0 ∞
x is positive on the right of the 0, negative on the left of the 0. x+1 is positive on right of the 1, and negative on the left of the 1. If we multiply the signs, we get the signs for the function. We are interested in the positive sign (because we need it to be greater than 0), so the interval is:
Whenever we have a fraction, we have to make a table:


x  –  –  + 
x+1  –  +  + 
+  –  + 
x is positive on the right of the 0, negative on the left of the 0. x+1 is positive on right of the 1, and negative on the left of the 1. If we multiply the signs, we get the signs for the function. We are interested in the positive sign (because we need it to be greater than 0), so the interval is:
(∞, 1) U (0, +∞)
Taking a Test? We can Help!
Practice Questions  Online Study Practice Courses  How to Answer Multiple ChoiceInequality Practice Questions
 Solve the inequality:
7x – 1 ≥ 13
1) [2 +∞)
2) (7, +∞)
3) (∞, 2]
4) (2, +∞)
 Solve the inequality:
2x – 1 ≥ x + 10
1) (∞, 9)
2) (9, +∞)
3) (∞, 9]
4) [11, +∞)
3. Solve the inequality:
(x – 6)^{2} ≥ x^{2} + 12
1) [2, +∞)
2) (2, +∞)
3) (∞, 2]
4) (12, +∞)
Taking a Test? We can Help!
Practice Questions  Online Study Practice Courses  How to Answer Multiple ChoiceAnswer Key
1. 3) (∞, 2]
7x – 1 > 13
7x > 13 + 1
7x > 14
x > 2/(1)
x < 2
2. 4) [11, +∞)
2x – 1 > x + 10
2x – x > 10 + 1
x > 11
3. 3) (∞, 2]
(x – 6)^{2} > x^{2} + 12
x^{2} – 12x + 36 > x^{2} + 12
12x > 12 – 36
12x > 24
x > 2/(1)
x < 2
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4 Comments
I am a bit confused on a problem. For number three of the inequality practice problem.
3. 3) (∞, 2]
(x – 6)2 > x2 + 12
x2 – 12x + 36 > x2 + 12 where did the 12x come from?
the 12x is from (x – 6)2 – which is (x – 6)(x – 6)
There is a typo in the first example where x > 13. You have the interval set as (3, +∞), shouldn’t it be (13, +∞)..
got it thanks!