How to Solve Linear Inequalities?

Q: What is a linear inequality?

A linear inequality compares two linear expressions using , ≤ or ≥ instead of an equals sign, such as 2x + 3 > 7. Instead of one answer, it has a whole range of solutions. Here, subtracting 3 and dividing by 2 gives x > 2, so 2.5, 3, 10 and every other number greater than 2 make the statement true.

Q: How do you solve linear inequalities on a number line?

Solve the inequality first, then graph the answer. Place a circle at the boundary number: open for , closed (filled in) for ≤ or ≥. Shade the side that holds the solutions, right for greater than and left for less than. So x > 5 is an open circle at 5 shaded right, and x ≤ −3 is a closed circle at −3 shaded left.

Algebra 1

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.

Tutor-style math help

Solve Linear Inequalities: what to notice and how to work it

Inequalities skill

Inequalities describe a set of possible values. Solve the boundary like an equation, then decide which side of the boundary makes the statement true.

What to notice first

Watch the comparison sign from the first line to the last. Multiplying or dividing by a negative reverses the direction.

Common student mistake

Do not forget open and closed endpoints. Strict signs use open circles; signs with equals use closed circles.

Key formulas and cues

$a<b$

$a\le b$

$\text{multiply/divide by a negative} \Rightarrow \text{reverse the sign}$

$|x-a|<b \Rightarrow a-b<x<a+b$

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

Example: $-3x>12$

Divide both sides by -3.
Reverse the inequality sign.
Simplify 12 divided by -3.

Answer: $x<-4$

Keep the sign

Example: $x+5\le9$

Subtract 5 from both sides.
No negative multiplication or division happened.
Keep the sign direction.

Answer: $x\le4$

Open pop-up practice

Try one before moving on

Try: Solve $-2x\le10$.

Answer: $x\ge-5$. Divide by -2 and flip the sign.

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

Step-by-Step Solver Worksheet Maker ✏Practice Drills Flashcards

Illustration of students learning Linear Inequalities

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 $x < 4$.

The big idea

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.

Tutor tip: Open vs. Closed circle tracks the symbol, strict ($<$, $>$) is open (endpoint not included); inclusive ($\le$, $\ge$) is closed (endpoint included).

Picture the solution

$2x – 3 < 5$ → $x < 4$

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 inequality

Worked 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)

$\color{blue}{x>4}$
$\color{blue}{x\le 4}$
$\color{blue}{x<-4 \text{ (flipped)}}$
$\color{blue}{x\le 3 \text{ (flipped)}}$

1. $x > 4$

2. $x \le 4$

3. $x < -4$

4. $x \le 3$

Keep practicing

Make Your Own Inequalities Worksheet

Generate fresh linear inequalities with a full answer key, print or save as a PDF.

New problems every click, never the same sheet twice

Step-by-step answer key so you can self-check

Open the Worksheet Maker Inequality Solver

Frequently Asked Questions

What is a linear inequality?

A linear inequality compares two linear expressions using <, >, ≤ or ≥ instead of an equals sign, such as 2x + 3 > 7. Instead of one answer, it has a whole range of solutions. Here, subtracting 3 and dividing by 2 gives x > 2, so 2.5, 3, 10 and every other number greater than 2 make the statement true.

How do you solve linear inequalities step by step?

Simplify each side, move the variable terms to one side and the numbers to the other, then divide by the coefficient. For 3x − 7 > 8, add 7 to get 3x > 15, then divide by 3 to get x > 5. If you divide or multiply by a negative, flip the sign: 5 − 2x ≥ 11 becomes −2x ≥ 6, so x ≤ −3.

How do you solve linear inequalities with fractions?

Clear the fractions first by multiplying every term by the least common denominator. Because the LCD is positive, the inequality sign stays the same. For x/4 + 1/2 < 3, multiply each term by 4 to get x + 2 < 12, then subtract 2 to get x < 10. Clearing fractions early avoids most arithmetic slips.

How do you solve linear inequalities on a number line?

Solve the inequality first, then graph the answer. Place a circle at the boundary number: open for < or >, closed (filled in) for ≤ or ≥. Shade the side that holds the solutions, right for greater than and left for less than. So x > 5 is an open circle at 5 shaded right, and x ≤ −3 is a closed circle at −3 shaded left.

How do you solve linear inequality word problems?

Choose a variable, translate phrases like “at most” (≤) and “at least” (≥), then solve as usual. Example: you have $40 and movie tickets cost $6 each plus a $4 booking fee. That gives 6t + 4 ≤ 40, so 6t ≤ 36 and t ≤ 6. You can buy at most 6 tickets. Always check that the answer makes sense in the story.

How do you solve linear inequalities with two signs?

An inequality with two signs, like −3 < 2x + 1 ≤ 9, is a compound inequality. Do the same step to all three parts at once. Subtract 1 from each part to get −4 < 2x ≤ 8, then divide each part by 2 to get −2 < x ≤ 4. On a number line, that is an open circle at −2, a closed circle at 4 and shading between them.

Continue Your Study

Ready for the next step? Pick up right where this lesson leaves off:

Up nextHow to Solve Systems of EquationsContinue with Inequalities and Systems of EquationsOpen lesson →Up nextSystems of Equations with SubstitutionContinue with Inequalities and Systems of EquationsOpen lesson →Up nextHow to Graph Quadratic FunctionsContinue with Inequalities and Systems of EquationsOpen lesson →

Effortless Math Team

2 hours ago

Effortless Math Team

Related to This Article

Linear Inequalities

What people say about "How to Solve Linear Inequalities (Flip the Sign Rule)"?

No one replied yet.

How to Solve Linear Inequalities?

Linear Inequalities