Solving Linear Inequalities
These examples illustrate the properties that we use for solving inequalities.
Properties of Inequality
Addition Property of Inequality
If the same number is added to both sides of an inequality, then the solution
set to the inequality is unchanged.
Multiplication Property of Inequality
If both sides of an inequality are multiplied by the same positive number,
then the solution set to the inequality is unchanged.
If both sides of an inequality are multiplied by the same negative number and
the inequality symbol is reversed, then the solution set to the inequality is
unchanged.
Because subtraction is defined in terms of addition, the addition property of
inequality also allows us to subtract the same number from both sides. Because
division is defined in terms of multiplication, the multiplication property of
inequality also allows us to divide both sides by the same nonzero number as
long as we reverse the inequality symbol when dividing by a negative number.
Equivalent inequalities are inequalities with the same solution set.
We find the solution to a linear inequality by using the properties to convert
it into an equivalent inequality with an obvious solution set, just as we do
when solving equations.
Example 1
Solving inequalities
Solve each inequality. State and graph the solution set.
a) 2x  7 < 1
b) 5  3x < 11
Solution
a) We proceed exactly as we do when solving equations:
2x  7 
< 1 
Original inequality 
2x 
< 6 
Add 7 to each side 
x 
< 3 
Divide each side by 2. 
The solution set is written in set notation as {x
 x < 3} and in interval notation as (∞,
3). The graph is shown below:
b) We divide by a negative number to solve this inequality.
5  3x 
< 11 
Original equation 
3x 
< 6 
Subtract 5 from each side 
x 
> 2 
Divide each side by 3 and reverse the inequality symbol 
The solution set is written in set notation as {x
 x > 2} and
in interval notation as (2, ∞). The graph is shown below:
Example 2
Solving inequalities
Solve
State and graph the solution set.

≥ 4 
Original inequality 

≤ 5(4) 
Multiply each side by 5 and reverse the inequality symbol. 
8 + 3x 
≤ 20 
Simplify 
3x 
≤ 12 
Subtract 8 from each side. 
x 
≤ 4 
Divide each side by 3. 
The solution set is (∞, 4], and its graph is
shown below:
