Multi-Step Inequalities
Multi-step inequalities are solved like multi-step equations: distribute, combine like terms, and get the variable terms on one side. Then undo adding or subtracting, and multiplying or dividing. Flip the sign only if you multiply or divide by a negative number. For example, 3(x - 2) + x \gt 10 gives 4x \gt 16, so x \gt 4.
Before this lesson: You should know how to solve two-step inequalities and multi-step equations.
Extra practice: The Algebra 1 Unit 2 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.
What Is a Multi-Step Inequality?
A multi-step inequality has parentheses, like terms, or the variable on both sides. For example, 5a - 3 \ge 2a + 9 has an a-term on each side.
The plan is the same as for equations: simplify first, then solve. The only extra rule is the one you already know. Flip the sign when you multiply or divide by a negative.
How to Solve Multi-Step Inequalities
- Distribute to remove parentheses.
- Combine like terms on each side.
- If the variable is on both sides, add or subtract to move all variable terms to one side.
- Undo adding or subtracting.
- Undo multiplying or dividing. Flip the sign if the number is negative.
- Graph the answer and test a point.
Adding or subtracting a term, even a variable term like 2x, never flips the sign. Only multiplying or dividing by a negative number does.
Worked Examples
Example 1: Distribute and combine
Solve 3(x - 2) + x \gt 10.
3x - 6 + x \gt 10
Step 1. Distribute the 3.
4x - 6 \gt 10
Step 2. Combine like terms (see combining like terms).
4x \gt 16
Step 3. Add 6 to both sides.
x \gt 4
Step 4. Divide both sides by 4. Four is positive, so don’t flip.
Answer: x \gt 4
Check: Test 5: 3(5 - 2) + 5 = 9 + 5 = 14, and 14 \gt 10. True.
Example 2: Distribute a negative
Solve -2(y + 1) \le 8.
-2y - 2 \le 8
Step 1. Distribute -2 to both terms.
-2y \le 10
Step 2. Add 2 to both sides.
y \ge -5
Step 3. Divide both sides by -2. It’s negative, so flip the sign (see one-step inequalities).
Step 4. Graph it: closed circle at -5, shade right, as in the graph above.
Answer: y \ge -5
Check: Test 0: -2(0 + 1) = -2, and -2 \le 8. True.
Example 3: Variables on both sides
Solve 5a - 3 \ge 2a + 9.
3a - 3 \ge 9
Step 1. Subtract 2a from both sides to get the variable on one side. The sign doesn’t change.
3a \ge 12
Step 2. Add 3 to both sides.
a \ge 4
Step 3. Divide both sides by 3.
Answer: a \ge 4
Check: Test 4: 5(4) - 3 = 17 and 2(4) + 9 = 17, and 17 \ge 17. True.
Example 4: Parentheses and variables on both sides
Solve 4 - 3(k - 2) \lt 2k.
4 - 3k + 6 \lt 2k
Step 1. Distribute -3. A negative times a negative is positive, so (-3)(-2) = 6.
10 - 3k \lt 2k
Step 2. Combine the constants on the left.
10 \lt 5k
Step 3. Add 3k to both sides. Moving the variable to the side where it stays positive avoids dividing by a negative.
2 \lt k
Step 4. Divide both sides by 5.
k \gt 2
Step 5. Write the variable first. The sign turns around with it.
Answer: k \gt 2
Check: Test 3: 4 - 3(3 - 2) = 4 - 3 = 1, and 1 \lt 2(3). True.
Example 5: Word problem
Phone plan A costs $30 a month plus $0.10 per text. Plan B costs $15 a month plus $0.25 per text. For how many texts is plan A cheaper?
30 + 0.10t \lt 15 + 0.25t
Step 1. Let t = the number of texts. Plan A is cheaper when its cost is less than plan B’s cost.
30 \lt 15 + 0.15t
Step 2. Subtract 0.10t from both sides.
15 \lt 0.15t
Step 3. Subtract 15 from both sides.
100 \lt t
Step 4. Divide both sides by 0.15 (see operations with decimals).
t \gt 100
Step 5. Write the variable first.
Answer: Plan A is cheaper for more than 100 texts a month.
Check: Try 200 texts. Plan A costs 30 + 0.10(200) = 30 + 20 = 50 dollars and plan B costs 15 + 0.25(200) = 15 + 50 = 65 dollars.
Common Mistakes
| Mistake | What it looks like | How to fix it |
|---|---|---|
| Distributing a negative wrong | -2(y + 1) becomes -2y + 1 | Multiply both terms by -2: -2y - 2. |
| Flipping when you subtract a variable term | 5a - 3 \ge 2a + 9 becomes 3a - 3 \le 9 | Subtracting 2a doesn’t flip the sign. Keep greater than or equal to. |
| Forgetting to flip | -2y \le 10 becomes y \le -5 | You divided by -2: y \ge -5. |
| Misreading the final answer | 10 \lt 5k becomes k \lt 2 | 2 \lt k means k \gt 2. Rewrite with the variable first. |
| Stopping when the variables cancel | 2(x + 3) \gt 2x + 1 gives 6 \gt 1, so “no answer” | A true statement means every number works. See the next section. |
When the Variables Cancel
Sometimes the variable terms subtract away completely. Then look at what’s left:
\begin{aligned}
2(x + 3) &\gt 2x + 1 & &\Rightarrow & 6 &\gt 1 & &\text{always true: all real numbers} \\
2x + 5 &\lt 2(x + 1) & &\Rightarrow & 5 &\lt 2 & &\text{never true: no solution}
\end{aligned}If the statement left over is true, every number is a solution. If it’s false, no number works. The same idea shows up in equations with variables on both sides.
Practice Problems
Solve each inequality.
- 2(x + 4) \gt 14
- 5 - (y - 3) \le 2
- 3m + 4 - m \lt 18
- -4(n - 2) \ge 20
- 7x - 5 \gt 3x + 11
- 2(p + 1) \le 5p - 7
- 6 - 2w \ge 3(w + 7)
- \frac{1}{2}(4x - 6) \lt x + 5
- 3(x - 1) \ge 3x - 5
- Gym A charges $40 to join and $30 a month. Gym B charges $100 to join and $20 a month. For how many months is gym A cheaper?
Answers
Show answers
1.
2x + 8 \gt 14
Distribute the 2.
2x \gt 6
Subtract 8 from both sides.
x \gt 3
Divide both sides by 2.
2.
5 - y + 3 \le 2
Distribute the −1.
8 - y \le 2
Combine the constants.
-y \le -6
Subtract 8 from both sides.
y \ge 6
Divide both sides by −1 and flip the sign.
3.
2m + 4 \lt 18
Combine like terms.
2m \lt 14
Subtract 4 from both sides.
m \lt 7
Divide both sides by 2.
4.
-4n + 8 \ge 20
Distribute the −4.
-4n \ge 12
Subtract 8 from both sides.
n \le -3
Divide both sides by −4 and flip the sign.
5.
4x - 5 \gt 11
Subtract 3x from both sides.
4x \gt 16
Add 5 to both sides.
x \gt 4
Divide both sides by 4.
6.
2p + 2 \le 5p - 7
Distribute the 2.
2 \le 3p - 7
Subtract 2p from both sides.
9 \le 3p
Add 7 to both sides.
3 \le p
Divide both sides by 3.
p \ge 3
Write the variable first.
7.
6 - 2w \ge 3w + 21
Distribute the 3.
6 - 5w \ge 21
Subtract 3w from both sides.
-5w \ge 15
Subtract 6 from both sides.
w \le -3
Divide both sides by −5 and flip the sign.
8.
2x - 3 \lt x + 5
Distribute the 1/2.
x - 3 \lt 5
Subtract x from both sides.
x \lt 8
Add 3 to both sides.
9.
3x - 3 \ge 3x - 5
Distribute the 3.
-3 \ge -5
Subtract 3x from both sides.
That is always true, so the answer is all real numbers.
10.
40 + 30m \lt 100 + 20m
Let m = the number of months. Gym A is cheaper when its cost is less.
40 + 10m \lt 100
Subtract 20m from both sides.
10m \lt 60
Subtract 40 from both sides.
m \lt 6
Divide both sides by 10.
Gym A is cheaper for fewer than 6 months.
Want more practice? The Algebra 1 Unit 2 Practice Pack has 51 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.
Multi-Step Inequalities FAQ
What is a multi-step inequality?
It is an inequality that takes more than two steps to solve, usually because of parentheses, like terms, or variables on both sides.
How do you solve multi-step inequalities?
Simplify each side, move the variable terms to one side, then undo adding or subtracting and multiplying or dividing. Flip the sign if you multiply or divide by a negative.
Do you flip the sign when you move a variable to the other side?
No. Moving a term means adding or subtracting it, and that never flips the sign. For example, subtracting 2a from both sides keeps the same sign.
What does it mean if the variables cancel out?
If what’s left is true, like 6 \gt 1, every real number is a solution. If it’s false, like 5 \lt 2, there is no solution.
Summary
- Distribute and combine like terms first.
- Move the variable terms to one side by adding or subtracting. The sign doesn’t change.
- Flip the sign only when you multiply or divide by a negative number.
- Moving the variable to the side where its coefficient stays positive avoids a flip.
- If the variables cancel, a true statement means all real numbers and a false one means no solution.
Previous lesson: Two-Step Inequalities
Next lesson: Compound Inequalities
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