What is the difference between solving a linear equation like 2x + 3 = 7 and a linear inequality like 2x + 3 < 7?

The solving steps are identical—add and divide the same way—but the results differ. An equation gives one value (x = 2), while an inequality gives a range (x < 2). The sign-flip rule only applies to inequalities; equations never flip their equals sign.

Do I flip the sign if I subtract a negative number, like x - (-5)?

No. Only multiplication and division by negatives trigger the sign flip. Subtracting or adding never does. Rewrite x - (-5) as x + 5 first, then solve normally.

Can I have both x -2 as part of the same solution?

Yes, but that is called a compound inequality, not a single linear inequality. If both conditions must be true at the same time, write it as -2 < x < 3. A single linear inequality gives you one inequality symbol pointing one direction.

Linear Inequalities

Q: Why do you flip the inequality sign when multiplying or dividing by a negative number?

Negative numbers reverse order on a number line. For example, 3 > 2, but -3 < -2. When you multiply an inequality by a negative, the relationship must flip to stay true. This is not an arbitrary rule—it is a consequence of how negative numbers work.

Q: What is the difference between solving a linear equation like 2x + 3 = 7 and a linear inequality like 2x + 3 < 7?

The solving steps are identical—add and divide the same way—but the results differ. An equation gives one value (x = 2), while an inequality gives a range (x < 2). The sign-flip rule only applies to inequalities; equations never flip their equals sign.

Q: Do I flip the sign if I subtract a negative number, like x - (-5)?

No. Only multiplication and division by negatives trigger the sign flip. Subtracting or adding never does. Rewrite x - (-5) as x + 5 first, then solve normally.

Solve linear inequalities by isolating the variable, flipping the inequality sign when multiplying or dividing by a negative number.

ax + b < c, \quad a < 0 \implies \text{flip the sign}

Solve a problem with Linear Inequalities

Type the problem. The solver will use Linear Inequalities where Linear Inequalities is the right tool, and tell you when it is not.

What each symbol means

What Linear Inequalities takes

Symbol	Meaning
$a$	The coefficient of $x$ ; its sign (positive or negative) determines whether you flip the inequality sign when solving.
$b$	A constant added to $ax$ ; it shifts the expression but doesn't affect when to flip the sign.
$c$	The number on the right side of the inequality; solving means finding when $ax + b$ is less than (or greater than, etc.) this value.
$x$	The variable you're solving for; the answer will be a range like $x < 4$ rather than a single value.

When to use it

Use linear inequalities when you need to find a range of values that satisfy a condition, rather than a single answer.

Level

Usually taught in: Algebra I

Worked examples

1. Solve a simple inequality with a positive coefficient

Problem

Solve

2x - 3 < 5

for

x

$2x - 3 < 5$
Start with the inequality.
$2x < 8$
Add 3 to both sides to isolate the term with $x$ .
$x < 4$
Divide both sides by 2 (positive), so the inequality sign $<$ stays the same.

Answer: $x < 4$

This is a straightforward linear inequality where we use inverse operations to isolate $x$ , just like solving equations. Since we divide by a positive number, the inequality sign does not change. The answer means any number less than 4 satisfies the original inequality.

2. Solve an inequality with a negative coefficient

Problem

Solve

-3x + 4 \geq 10

for

x

$-3x + 4 \geq 10$
Start with the inequality.
$-3x \geq 6$
Subtract 4 from both sides to isolate the term with $x$ .
$x \leq -2$
Divide both sides by $-3$ (negative). The inequality sign flips from $\geq$ to $\leq$ .

Answer: $x \leq -2$

This example shows the critical rule: when you divide by a negative coefficient, you must flip the inequality sign. Without flipping, the answer would be wrong. The answer means $x$ can be any number less than or equal to $-2$ .

3. Apply an inequality to a word problem with time and scoring

Problem

In a speedrun game, if you take longer than 60 seconds, your score decreases by 12 points per extra second. Your score is given by

-12s + 300

, where

s

is the number of seconds over 60. For how many extra seconds will your score become negative?

$-12s + 300 < 0$
Set up the inequality to find when the score is negative.
$-12s < -300$
Subtract 300 from both sides to isolate the term with $s$ .
$s > 25$
Divide both sides by $-12$ (negative). The inequality sign flips from $<$ to $>$ .

Answer: $s > 25$

Your score becomes negative when $s > 25$ , meaning after more than 25 extra seconds (85 seconds total). This demonstrates why sign-flipping is essential in context: without it, you'd incorrectly conclude the score was negative at the start of the run.

Common mistakes

Where Linear Inequalities usually goes wrong

The mistake	Why it is wrong	The fix
A student writes: $-2x > 8$ becomes $x > -4$ .	When dividing by a negative number, you must flip the inequality sign; this solution forgot to do that.	$-2x > 8$ divided by $-2$ gives $x < -4$ (flip $>$ to $<$ ).
A student writes: $2x - 5 < 9$ becomes $2x < 14$ , and then $x < 7$ becomes $x > 7$ .	The sign only flips when you multiply or divide by a negative number; dividing by 2 (positive) does not flip the sign.	$2x < 14$ divided by 2 is $x < 7$ (keep the $<$ ).
A student writes: To solve $-x + 5 > 12$ , first get $-x > 7$ , so $x > -7$ .	After isolating to $-x > 7$ , dividing by the negative coefficient $-1$ requires flipping the sign.	$-x > 7$ becomes $x < -7$ (flip $>$ to $<$ because you divided by $-1$ ).

Tips and when to use something else

Always flip the inequality sign when multiplying or dividing by any negative number—check the sign of the coefficient before each operation.
Linear inequalities follow the same steps as equations (add, subtract, multiply, divide) until the sign-flip rule applies; think of them as equations first.
Graph your answer on a number line to check that it makes intuitive sense and aligns with the original problem.
If the problem involves absolute values like $|2x - 3| < 5$ , use Absolute Value Inequalities instead—that method handles both cases for you.

Related formulas

Linear EquationSolve for the value of x in any linear equation by rearranging to isolate x, finding the unique solution where the expression equals zero.Two-Step EquationsSolve two-step equations of the form ax + b = c by undoing operations in reverse order to isolate the variable using inverse operations.Compound InequalitiesCompound inequalities describe a range where a variable falls between two values; use them when bounds constrain something from both above and below.Absolute Value InequalitiesAbsolute value inequalities express ranges of values at a fixed distance from a point. Use them to solve distance, tolerance, or deviation problems.

Frequently asked questions

Why do you flip the inequality sign when multiplying or dividing by a negative number?: Negative numbers reverse order on a number line. For example, $3 > 2$ , but $-3 < -2$ . When you multiply an inequality by a negative, the relationship must flip to stay true. This is not an arbitrary rule—it is a consequence of how negative numbers work.
What is the difference between solving a linear equation like $2x + 3 = 7$ and a linear inequality like $2x + 3 < 7$ ?: The solving steps are identical—add and divide the same way—but the results differ. An equation gives one value ( $x = 2$ ), while an inequality gives a range ( $x < 2$ ). The sign-flip rule only applies to inequalities; equations never flip their equals sign.
Do I flip the sign if I subtract a negative number, like $x - (-5)$ ?: No. Only multiplication and division by negatives trigger the sign flip. Subtracting or adding never does. Rewrite $x - (-5)$ as $x + 5$ first, then solve normally.
Can I have both $x < 3$ and $x > -2$ as part of the same solution?: Yes, but that is called a compound inequality, not a single linear inequality. If both conditions must be true at the same time, write it as $-2 < x < 3$ . A single linear inequality gives you one inequality symbol pointing one direction.

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Reviewed 2026-09-18