How to Solve Multi-Step Inequalities

Q: Do I flip the sign every time I see a negative number?

No. You flip only when you multiply or divide BOTH sides by a negative. Subtracting a negative, or ending up with a negative answer, changes nothing about the symbol.

Q: How do I check an inequality answer?

Pick any number from inside your solution set and substitute it into the original inequality; it should be true. Then pick a number outside and check that it is false. Two tests catch almost every sign error.

Q: What if the variable ends up on the right?

That is fine, but read it carefully. 5 5 says the same thing. The symbol always points at the smaller side.

Quick answer

Solve a multi-step inequality the same way as a multi-step equation: distribute, combine like terms, then undo operations in reverse order. Flip the inequality sign only when you multiply or divide both sides by a negative number. For 3x - 7 > 8, add 7 to get 3x > 15, then divide by 3 to get x > 5.

What you'll learn

Solve inequalities that need more than one step
Decide correctly when the inequality sign must flip
Check a solution set by testing a value from inside it

The method you already know

A multi-step inequality is solved with the same sequence as a multi-step equation:

Distribute to clear parentheses.
Combine like terms on each side.
Move the variable terms to one side.
Undo addition or subtraction.
Undo multiplication or division — flip the sign if you divide by a negative.
Check with a value from inside your solution set.

Only step 5 differs from solving an equation, and only in the one case covered in one-step inequalities.

Why only the last step can flip

It helps to know exactly where the danger is, so you are not anxious about it at every line.

Steps 1 to 4 involve adding, subtracting, and distributing. None of those reverses the order of the number line — they slide values along it or rewrite one side into an equal form. Order is untouched, so the symbol is untouched.

Only multiplying or dividing by a negative reflects the number line, and that is what reverses order. So there is exactly one moment to watch: when you divide by the coefficient, look at its sign.

Worked examples

Common mistakes

Practice problems

Solve $2x + 5 > 17$ .

Hint

Subtract first, then divide. The coefficient is positive.

Answer

$x > 6$

Full solution

Subtract $5$ : $2x > 12$ . Divide by $2$ : $x > 6$ .

Check: $x = 7$ gives $19 > 17$ ✓; $x = 6$ gives $17 > 17$ , false, so the boundary is correctly excluded.
Solve $5y - 3 \le 12$ .

Answer

$y \le 3$

Full solution

Add $3$ : $5y \le 15$ . Divide by $5$ : $y \le 3$ .

Check: $y = 3$ gives $12 \le 12$ ✓.
Solve $-3x + 4 > 19$ .

Hint

Isolate $-3x$ first, then watch the sign of the number you divide by.

Answer

$x < -5$

Full solution

Subtract $4$ : $-3x > 15$ . Divide by $-3$ and flip: $x < -5$ .

Check: $x = -6$ gives $-3(-6) + 4 = 22 > 19$ ✓, and $-6 < -5$ ✓.
Solve $4(x + 2) \ge 20$ .

Answer

$x \ge 3$

Full solution

Divide both sides by $4$ : $x + 2 \ge 5$ . Subtract $2$ : $x \ge 3$ .

Check: $x = 3$ gives $4(5) = 20 \ge 20$ ✓.
Solve $7 - 2x < 1$ .

Answer

$x > 3$

Full solution

Subtract $7$ : $-2x < -6$ . Divide by $-2$ and flip: $x > 3$ .

Check: $x = 4$ gives $7 - 8 = -1 < 1$ ✓.
Solve $3x + 2 \le 5x - 8$ .

Hint

Move the variables to the side that keeps the coefficient positive.

Answer

$x \ge 5$

Full solution

Subtract $3x$ : $2 \le 2x - 8$ . Add $8$ : $10 \le 2x$ . Divide by $2$ : $5 \le x$ , that is $x \ge 5$ .

Check: $x = 5$ gives $17 \le 17$ ✓; $x = 4$ gives $14 \le 12$ , false ✓.
Solve $-(x - 6) \ge 2x$ .

Hint

The minus in front distributes as $-1$ across both terms.

Answer

$x \le 2$

Full solution

Distribute: $-x + 6 \ge 2x$ . Add $x$ to both sides: $6 \ge 3x$ . Divide by $3$ : $2 \ge x$ , that is $x \le 2$ .

Check: $x = 2$ gives $-(2-6) = 4$ and $2(2) = 4$ , so $4 \ge 4$ ✓.
A phone plan costs $30 per month plus $0.10 per text. You want to spend no more than $45 in a month. How many texts can you send?

Hint

“No more than” means the total is allowed to equal $45.

Answer

At most 150 texts.

Full solution

Let $t$ be the number of texts: $0.10t + 30 \le 45$ .

Subtract $30$ : $0.10t \le 15$ . Divide by $0.10$ : $t \le 150$ .

Since $t$ counts texts it must be a whole number, so the answer is at most $150$ texts.

Check: $150$ texts cost $0.10(150) + 30 = 45 \le 45$ ✓.

Frequently asked questions

Do I flip the sign every time I see a negative number?

No. You flip only when you multiply or divide BOTH sides by a negative. Subtracting a negative, or ending up with a negative answer, changes nothing about the symbol.

How do I check an inequality answer?

Pick any number from inside your solution set and substitute it into the original inequality; it should be true. Then pick a number outside and check that it is false. Two tests catch almost every sign error.

What if the variable ends up on the right?

That is fine, but read it carefully. 5 < x means x is greater than 5, so writing it as x > 5 says the same thing. The symbol always points at the smaller side.

What to learn next

How to Solve Compound Inequalities (AND and OR)Solve compound inequalities joined by AND or OR, graph the solution on a number line, and tell which combinations give no solution or every number.

Key terms in this lesson

Coefficient: A coefficient is the number multiplying a variable in a term. In 3x the coefficient is 3, and in -5y it is -5, because the sign belongs to the number.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.HSA.REI.B.3Reasoning with Equations and InequalitiesSolve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

For teachersPrint problems onlyTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.

Quick answer

What you'll learn

The method you already know

Why only the last step can flip

Worked examples

Common mistakes

Practice problems

Frequently asked questions

Do I flip the sign every time I see a negative number?

How do I check an inequality answer?

What if the variable ends up on the right?

What to learn next

Key terms in this lesson

Standards alignment

More Algebra 1 lessons