A linear inequality is an algebraic statement that contains >, < , ≥ , or ,
such as y ≥ 2x + 7 or 3x + 2y − z < 4.
Things you can do to solve linear inequalities:
You can add or subtract the same term from both sides of an inequality.
Example:
-y + 5 < 17
-y + y < 17 + y (add +y to both sides)
-y + y = 0 < 17 + y
0 - 17 < 17 - 17 +y (subtract 17 from both sides)
y < - 17
You can multiply both sides of an equation by the same term; but remember, when you multiply each side, you must multiply each term in that side by the multiplier. You can also divide both sides of an equation by the same term, but you also must divide each term in that side by the dividing term.
You may also have to solve an inequality by multiple steps, such as multiplying and then subtracting.
Example:
(5x - 2)/ 6 > 4 + 3x
5x - 2 > 6(4 +3x) > 24 + 18x multiply both sides by 6
- 2 > 24 +18x - 5x subtract 5x from both sides
- 2 - 24 > 13x subtract 24 from both sides
x < - 26/13 divide both sides by 13
x < - 2
Remember, - 26/13 > x is the same as x < - 26/13
The inequality sign remains the same when you multiply or divide by a positive number.
The inequality sign is reversed when you multiply or divide by a negative number.
- 6 x > 42
x < 42/ - 6 ( Dividing both sides by - 6, reverse > to < )
x < - 7
Solving Compound Inequalities
When two inequalities are considered together, they form a compound inequality. A compound inequality can be written two ways, one containing and; one containing or.
A compound inequality that contains an and is true only if both inequalities are true.
The graph of a compound inequality containing an and is the intersection of the graphs of the two inequalities:
Graph A is the inequality x ≤ 9. Graph B is the inequality x > 6.
The intersection of these two graphs is: 6 < x ≤ 9. Notice, the solution does not include the point 6, since 6 is not part of the inequality x > 6.
A compound inequality that contains an or is true only if one or more of the inequalities is true.
Graph A is the inequality: x ≤ 0 Graph B is x > 6.
The solution to the compound inequality of thee two is:
x ≤ 0 or x > 6.
Absolute Value Inequalities
To solve inequalities in the form of | x | < n, or | x | ≤ n, find the intersection of the two inequalities: x < n, and x > - n. Treat it as a compound inequality containing an and. Remember, for the case - x < n, we change the < to > when we multiply both sides by - 1. So we get x > - n.
Example:
| a + 7| < 8
Case 1: a + 7 < 8; therefore a < 1
Case 2: a + 7 > - 8; therefore a > - 15
The solution is: - 15 < a < - 1
To solve inequalities in the form of | x | > n, or | x | ≥ n, find the union of the two inequalities: x < n, and x > - n. Treat it as a compound inequality containing an or. Remember, for the case - x < n, we change the < to > when we multiply both sides by - 1. So we get x > - n.
Example:
| x - 5| > 12
Case1: x - 5 > 12; therefore x > 17
Case 2: x - 5 < - 12; therefore x < - 7
The solution is: x > 17 or x < -7
Inequalities with 2 variables
The graph of the inequality y ≤ 1.25 x + 1.25 is above. Note that the graph includes the line represented by y = 1.25 x + 1.25, and all points below this line.
