6.3: MultiStep Inequalities
What if you had an inequality with an unknown variable on both sides like
Watch This
CK12 Foundation: 0603S Solving MultiStep Inequalities (H264)
Try This
For additional practice solving inequalities, try the online game at http://www.aaamath.com/equ725x7.htm#section2. If you’re having a hard time with multistep inequalities, the video at http://www.schooltube.com/video/aa66df49e0af4f85a5e9/MultiStepInequalities will walk you through a few.
Guidance
In the last two sections, we considered very simple inequalities which required one step to obtain the solution. However, most inequalities require several steps to arrive at the solution. As with solving equations, we must use the order of operations to find the correct solution. In addition, remember that when we multiply or divide the inequality by a negative number, the direction of the inequality changes.
The general procedure for solving multistep inequalities is almost exactly like the procedure for solving multistep equations:
 Clear parentheses on both sides of the inequality and collect like terms.
 Add or subtract terms so the variable is on one side and the constant is on the other side of the inequality sign.
 Multiply and divide by whatever constants are attached to the variable. Remember to change the direction of the inequality if you multiply or divide by a negative number.
Example A
Solve the inequality
Solution
Original problem:
Add
Simplify:
Add 7 to both sides:
Simplify:
Multiply 5 to both sides:
Simplify:
Divide both sides by 24:
Simplify:
Graph:
Example B
Solve the inequality
Solution:
Original problem:
Add
Simplify:
Subtract 12:
Simplify:
Divide both sides by 15:
Simplify:
Graph:
Example C
Solve the inequality
Solution:
Original problem:
Simplify parentheses:
Collect like terms:
Add
Simplify:
Subtract 18:
Simplify:
Divide both sides by 6:
Simplify:
Watch this video for help with the Examples above.
CK12 Foundation: Solving MultiStep Inequalities
Vocabulary
 The answer to an inequality is usually an interval of values.
 Solving inequalities works just like solving an equation. To solve, we isolate the variable on one side of the equation.
 When multiplying or dividing both sides of an inequality by a negative number, you need to reverse the inequality.
Guided Practice
Solve the inequality
Solution:
Original problem:
Simplify parenthesis:
Multiply both sides by 4:
Simplify:
Add
Simplify:
Add 1 to both sides:
Simplify:
Divide both sides by 13:
Simplify:
Practice
Solve each multistep inequality.

3x−5<x+3 
x−5>2x+3 
2(x−3)≤3x−2 
3(x+1)≥2x+5 
2(x−9)≥−1(4x+7) 
x3<x+7  \begin{align*}\frac{x}{4} < 2x21\end{align*}
 \begin{align*}\frac{3(x4)}{12} \le \frac{2x}{3}\end{align*}
 \begin{align*}2 \left ( \frac{x}{4} + 3\right ) > 6(x1)\end{align*}
 \begin{align*}9x+4 \le 2 \left ( x+\frac{1}{2} \right )\end{align*}
Notes/Highlights Having trouble? Report an issue.
Color  Highlighted Text  Notes  

Show More 
Image Attributions
Here you'll learn how to solve inequalities that require several steps to arrive at the solution. You'll also graph their solution set.