Solving Compound Linear Inequalities
A compound inequality contains two inequality symbols.
For example, to indicate all the numbers
between 4 and 6 we could write the following: 4 < x and x < 6
This says that 4 is less than x and x is also
less than 6.
We can say the same thing using this
compound inequality: 4 < x < 6
To solve a compound inequality, apply the same operations to the left side,
the middle part, and the right side. A compound inequality is solved when
the variable has been isolated in the middle part.
Example 1
Solve: 25 ≥ 1  4y > 1. Then, graph the solution
on a number line.
Solution 
25 ≥ 
1  4y 
> 1 
Subtract 1 from each part.
Simplify.

25  1 ≥
24 ≥ 
1  4y  1 4y 
> 1  1 > 0 
Divide each part by 4. Since we are
dividing by a negative, we must reverse
both of the inequality symbols. 
≤ 


Simplify. 
6 ≤

y 
< 0 
The compound inequality 6 ≤ y < 0 includes all real numbers
between 6 and 0, including 6.
To graph the solution, plot a closed circle at 6 and an open circle at 0.
Then, shade the region between 6 and 0.
Example 2
Solve: 3 + x < 4x  6 < 9 + x. Then, graph the solution on a
number line.
Solution
Subtract x from each part.
Simplify.
Add 6 to each part.
Simplify. 
3 + x < 3 + x  x <
3 <
3 + 6 <
9 < 
4x  6 4x  6  x
3x  6
3x  6 + 6
3x 
< 9 + x < 9 + x  x
< 9
< 9 + 6
< 15 
Divide each part by 3.




Simplify. 
3 < 
x 
< 5 
To graph the solution, place an open circle at 3 and an open circle at 5.
Then, shade the region between the plotted points.
