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.

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

1. Distribute to remove parentheses.
2. Combine like terms on each side.
3. If the variable is on both sides, add or subtract to move all variable terms to one side.
4. Undo adding or subtracting.
5. Undo multiplying or dividing. Flip the sign if the number is negative.
6. 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.

Step 1. Distribute the 3 and combine like terms.

Step 2. Add 6, then divide by 4.

\begin{aligned}
3(x - 2) + x &\gt 10 \\
3x - 6 + x &\gt 10 \\
4x - 6 &\gt 10 \\
4x &\gt 16 \\
x &\gt 4
\end{aligned}

Check: Test 5: 3(5 - 2) + 5 = 14, and 14 \gt 10. True.

Answer: x \gt 4

Example 2: Distribute a negative

Solve -2(y + 1) \le 8.

Step 1. Distribute -2: -2y - 2.

Step 2. Add 2, then divide by -2 and flip the sign.

\begin{aligned}
-2(y + 1) &\le 8 \\
-2y - 2 &\le 8 \\
-2y &\le 10 \\
y &\ge -5
\end{aligned}

Answer: y \ge -5. Closed circle at -5, shade right, as in the graph above.

Example 3: Variables on both sides

Solve 5a - 3 \ge 2a + 9.

Step 1. Subtract 2a from both sides to get the variable on one side. The sign doesn’t change.

Step 2. Add 3, then divide by 3.

\begin{aligned}
5a - 3 &\ge 2a + 9 \\
3a - 3 &\ge 9 \\
3a &\ge 12 \\
a &\ge 4
\end{aligned}

Answer: a \ge 4

Example 4: Parentheses and variables on both sides

Solve 4 - 3(k - 2) \lt 2k.

Step 1. Distribute -3 and combine: 4 - 3k + 6 = 10 - 3k.

Step 2. Add 3k to both sides. Moving the variable to the side where it stays positive avoids dividing by a negative.

\begin{aligned}
4 - 3(k - 2) &\lt 2k \\
10 - 3k &\lt 2k \\
10 &\lt 5k \\
2 &\lt k
\end{aligned}

Answer: 2 \lt k, which is k \gt 2.

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?

Step 1. Let t = the number of texts. Plan A is cheaper when its cost is less than plan B’s cost.

30 + 0.10t \lt 15 + 0.25t

Step 2. Subtract 0.10t and 15 from both sides, then divide by 0.15.

\begin{aligned}
15 &\lt 0.15t \\
100 &\lt t
\end{aligned}

Answer: Plan A is cheaper for more than 100 texts a month. Check with 200 texts: plan A costs 30 + 20 = 50 dollars and plan B costs 15 + 50 = 65 dollars.

Common Mistakes

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.

1. 2(x + 4) \gt 14
2. 5 - (y - 3) \le 2
3. 3m + 4 - m \lt 18
4. -4(n - 2) \ge 20
5. 7x - 5 \gt 3x + 11
6. 2(p + 1) \le 5p - 7
7. 6 - 2w \ge 3(w + 7)
8. \frac{1}{2}(4x - 6) \lt x + 5
9. 3(x - 1) \ge 3x - 5
10. 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

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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