How to Solve Linear Inequalities?
Linear Inequalities
Solving a linear inequality is like solving a linear equation, then remembering the one twist: dividing or multiplying by a negative flips the sign. The answer is a range of values you can picture on a number line. We’ll solve and graph several, with a solver, drills, and a worksheet maker a tap away.
Solve Linear Inequalities: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Solve the boundaryTemporarily treat the inequality like an equation.
- Choose the sideUse the sign or test a number if the direction is not obvious.
- Graph the solutionUse the correct endpoint and shade the values that work.
Worked examples
Flip the sign
- Divide both sides by -3.
- Reverse the inequality sign.
- Simplify 12 divided by -3.
Keep the sign
- Subtract 5 from both sides.
- No negative multiplication or division happened.
- Keep the sign direction.
Try one before moving on
Solve Linear Inequalities: pop-up practice
A linear inequality asks not “what value makes this true?” but “what range of values makes this true?” You solve it the same way you’d solve a linear equation, isolate the variable, and then graph the answer as a ray on a number line. There’s just one rule to respect: flipping the inequality whenever you multiply or divide by a negative.
In short: isolate the variable using inverse operations, flip the sign if you divide or multiply by a negative, and graph the result on a number line. For example, \(2x – 3 < 5\) gives \(x < 4\).
A Range, Not a Point
Where an equation pins the variable to one number, an inequality leaves a whole stretch of the number line true. Solving still means undoing operations to get the variable alone, and on a number line you mark the boundary (open circle for \(<\) or \(>\), closed for \(\le\) or \(\ge\)) and shade the direction that works.
How to solve (3 steps):
- Undo addition/subtraction, then multiplication/division to isolate the variable.
- Flip the inequality if you multiplied or divided by a negative.
- Graph the solution: circle the endpoint, shade the true side.
\(2x – 3 < 5\) → \(x < 4\)
Add 3: \(2x < 8\); divide by 2: \(x < 4\). On the number line that's an open circle at 4 with the arrow pointing left, every value less than 4 works.
Solve an inequalityWorked Examples
Example A, A two-step inequality
Solve \(2x – 3 < 5\).
- Add 3 to both sides: \(2x < 8\).
- Divide both sides by 2 (positive, so no flip): \(x < 4\).
- Graph it: open circle at 4 (strict), arrow pointing left.
Answer: \(x < 4\)
Example B, An inclusive inequality
Solve \(3x + 1 \ge 10\).
- Subtract 1 from both sides: \(3x \ge 9\).
- Divide both sides by 3: \(x \ge 3\).
- Graph it: closed circle at 3 (inclusive), arrow pointing right.
Answer: \(x \ge 3\)
Example C, Flip the sign
Solve \(-x + 2 \le 6\).
- Subtract 2 from both sides: \(-x \le 4\).
- Divide both sides by \(-1\), dividing by a negative, so flip: \(x \ge -4\).
- Graph it: closed circle at \(-4\), arrow pointing right.
Answer: \(x \ge -4\)
Example D, A negative coefficient
Solve \(4 – 2x > 10\).
- Subtract 4 from both sides: \(-2x > 6\).
- Divide both sides by \(-2\) and flip the sign: \(x < -3\).
- Graph it: open circle at \(-3\), arrow pointing left.
Answer: \(x < -3\)
Where You’ll Use It
Linear inequalities describe boundaries in real life: “spend no more than $50,” “score at least 90,” “stay under the weight limit.” Solving one converts the sentence into the exact set of allowed values, and the number-line picture makes that set easy to see at a glance.
Easy Points to Lose
- Forgetting the flip. Dividing or multiplying by a negative reverses the sign, always.
- Wrong circle. Use an open circle for \(<\)/\(>\) and a closed one for \(\le\)/\(\ge\).
- Shading the wrong way. After solving, read the final symbol: \(x > -3\) shades right, \(x < -3\) shades left.
- Treating it like an equation answer. The solution is a range, so a quick test value confirms you shaded correctly.
Your Turn: Solve and Graph
Solve each and note the direction, then reveal the answers.
- \(x + 5 > 9\)
- \(2x – 1 \le 7\)
- \(-3x > 12\)
- \(5 – x \ge 2\)
Show answers (with graphs)
- \(\color{blue}{x>4}\)
- \(\color{blue}{x\le 4}\)
- \(\color{blue}{x<-4 \text{ (flipped)}}\)
- \(\color{blue}{x\le 3 \text{ (flipped)}}\)
Make Your Own Inequalities Worksheet
Generate fresh linear inequalities with a full answer key, print or save as a PDF.
Frequently Asked Questions
How do I solve a linear inequality?
Isolate the variable with inverse operations, just like an equation, then flip the inequality sign if you multiplied or divided by a negative. Graph the resulting range on a number line.
When does the sign flip?
Only when you multiply or divide both sides by a negative number. Adding and subtracting never change the direction.
Open circle or closed circle?
Open for strict inequalities (\(<\), \(>\)) where the endpoint isn’t included; closed for \(\le\) and \(\ge\) where it is.
How do I check the solution?
Test a value from your shaded range in the original inequality, it should be true, and one outside it to confirm it’s false.
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Continue Your Study
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