A linear inequality is one that can be transformed into one of the standard forms:
The procedure for solving linear inequalities in one variable is similar to solving linear equations; apply transformation operation to isolate the variable on one side of the inequality symbol.ax + b < 0 ax + b ≤ 0 ax + b > 0 ax + b ≥ 0
Example, solve: x + 2 < 5
Then the solution set is the open interval (-∞, 3)x + 2 - 2 < 5 - 2 x < 3
Example, solve: 3 - 2x ≥ 15
Then the solution set is the half open interval (-∞, -6]3 - 2x - 3 ≥ 15 - 3 -2x ≥ 12 2x ≤ -12 2x/2 ≤ -12/2 x ≤ -6
This combined statement may be solved by isolating the variable in the center.a < b < c
Example, solve: 0 < 2x + 4 < 10
Then the solution set is the open interval (-2, 3)0 - 4 < 2x + 4 - 4 < 10 - 4 -4 < 2x < 6 -4/2 < 2x/2 < 6/2 -2 < x < 3
Isolation of the variable as above is not always possible. In this case, isolate the variables in two separate inequalities. The solution must satisfy both inequalities.
Example, solve: 3x + 8 > x - 4 ≥ 2x
Then the solution set is the half open interval (-6, -4]3x + 8 - x + 4 > x - 4 - x + 4 and x - 4 - 2x ≥ 2x - 2x 2x + 12 > 0 and -x - 4 ≥ 0 0 < x + 6 and x + 4 ≤ 0 -6 < x and x ≤ -4