How to Solve Linear Inequalities – Quick Review and Practice
- Posted by Brian Stocker
- Date March 19, 2019
- Comments 2 comments
How to Solve 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, +∞)
Linear Inequality 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 ≥ x2 + 12
1) [2, +∞)
2) (2, +∞)
3) (-∞, 2]
4) (12, +∞)
Answer 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 > x2 + 12
x2 – 12x + 36 > x2 + 12
-12x > 12 – 36
-12x > -24
-x > -2/(-1)
x < 2
Date Published: Tuesday, March 19th, 2019
Date Modified: Tuesday, February 22nd, 2022
Got a Question? Email me anytime - Brian@test-preparation.ca
2 Comments
the 12x is from (x – 6)2 – which is (x – 6)(x – 6)
got it thanks!