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, +∞)
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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} ≥ x^{2} + 12
1) [2, +∞)
2) (2, +∞)
3) (∞, 2]
4) (12, +∞)
Try a FREE Algebra Quiz
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} > x^{2} + 12
x^{2} – 12x + 36 > x^{2} + 12
12x > 12 – 36
12x > 24
x > 2/(1)
x < 2
Common Mistakes Answering Linear Equation Questions
 Misinterpreting the Question or Setting Up the Equation Incorrectly
 Failing to properly understand what the question is asking.
 Confusing linear equations with other types of algebra.
 Misreading the problem and setting up the wrong equation.
 Basic Arithmetic Errors
 Addition, subtraction, multiplication, or division errors. Basic Math Practice
 Incorrectly applying the order of operations (PEMDAS). Order of Operation Practice
 Solving for the Wrong Variable:
 Solving for the wrong variable.
 Sign Errors:
 Incorrectly using positive and negative signs.
 Failing to change the sign when performing basic operations.
 Fractions and Decimals:
 Incorrectly converting between fractions, decimals, and percentages. Factions, decimals and percent practice
 Incorrect Distribution
 Mistakes when applying the distributive property.
 Forgetting to distribute a negative sign through parentheses.
 Incorrectly Combining Like Terms
 Failing to properly combine like terms.
 Mistaking terms that are not alike for like terms.
 Skipping Steps:
 Skipping steps in the process, leading to errors in the solution.
 Not showing work, making it hard to identify where a mistake was made.
 Checking Solutions:
 Forgetting to plug the solution back into the original equation to check its correctness.
 Assuming an answer is correct without verification.
 Mistakes Graphing
 Errors plotting or connecting points.
 Misinterpreting the slope and intercept in the line equation.
 Time Management:
 Spending too much time on one problem and rushing through others.
 Not allocating enough time to review and check answers. Time Management on a Test
 Anxiety and Pressure:
 Feeling overwhelmed or stressed causing incorrect answers.
 Making careless mistakes due to test anxiety. Test Anxiety Tips
Date Published: Tuesday, March 19th, 2019
Date Modified: Tuesday, June 4th, 2024
Got a Question? Email me anytime  Brian@testpreparation.ca
2 Comments
the 12x is from (x – 6)2 – which is (x – 6)(x – 6)
got it thanks!