Linear Inequalities: Solving and the Sign-Flip Rule

A linear inequality is a comparison between two linear expressions, such as $3x - 4 \le 11$, where the goal is to find every value of the variable that makes the statement true. Unlike a linear equation, which has a single solution, a linear inequality describes a whole range of numbers along the number line.

What a Linear Inequality Is

A linear inequality uses one of four comparison symbols in place of the equals sign:

The variable appears only to the first power (no $x^2$, no $\sqrt{x}$, no $\frac{1}{x}$), which is what makes the inequality linear. A solution such as $x < 4$ is not one number but an interval: every value below $4$ works, including $3$, $0$, and $-100$, while $4$ itself does not.

You reach for linear inequalities whenever a problem sets a limit, a cutoff, or a constraint instead of an exact target: a budget you must stay under, a minimum score you need to pass, a safe temperature range, or the domain of a function.

The Sign-Flip Rule (and Why It Holds)

Solving a linear inequality uses the same moves as solving a linear equation, with one extra rule:

Adding or subtracting the same quantity, or multiplying and dividing by a positive number, never changes the direction. Only a negative factor flips it. To see why, watch a true statement on the number line:

On the number line $-2$ sits to the right of $-5$, so $-2 > -5$. Multiplying by $-1$ reflected both numbers across zero, which reversed their order. The comparison stays true only because the symbol turned from $<$ into $>$.

More generally, if $a < b$ and $c$ is negative, then $ca > cb$. The magnitudes scale the same way they would with a positive factor; it is purely the direction that reverses. Keeping the statement true is the whole point, so the flip is mandatory, not optional.

Worked Example 1: $3x - 4 \le 11$

Treat the first steps exactly as you would an equation. Add $4$ to both sides:

Now divide both sides by $3$. Because $3$ is positive, the sign does not change:

The solution set is every number up to and including $5$. On a number line you draw a closed dot at $5$ (because of the $\le$) and shade everything to the left:

<====●----+----+----+--->
          5

Check with a value inside the set, $x = 2$: $3(2) - 4 = 2$, and $2 \le 11$ is true. Check the boundary, $x = 5$: $3(5) - 4 = 11$, and $11 \le 11$ is true, confirming the closed endpoint.

Worked Example 2: $-2x + 3 > 11$ (the flip case)

Subtract $3$ from both sides — no flip, since this is subtraction:

Now divide both sides by $-2$. This is the moment the rule applies: dividing by a negative number reverses $>$ into $<$:

The solution is every number strictly less than $-4$. The dot at $-4$ is open (hollow) because the symbol is strict, and you shade to the left:

<====○----+----+----+--->
        -4

Test a value inside, $x = -5$: $-2(-5) + 3 = 13$, and $13 > 11$ is true. Test a value outside, $x = 0$: $-2(0) + 3 = 3$, and $3 > 11$ is false. Both tests confirm the boundary at $x < -4$ and prove the flip was necessary — without it you would have written the wrong half of the line.

Reading the Answer on a Number Line

The endpoint marker encodes the symbol, and the shading encodes the direction:

In interval notation, $x \le 5$ is written $(-\infty, 5]$ — a square bracket for "or equal to," a parenthesis for infinity and for strict inequalities. So $x < -4$ becomes $(-\infty, -4)$.

A Quick Two-Sided Example

Constraints often arrive as a chain, such as $-1 \le 2x + 3 < 9$. Operate on all three parts at once. Subtract $3$ everywhere:

Divide everything by $2$ (positive, so no flip):

The solution is the interval $[-2, 3)$: closed on the left, open on the right. If the middle coefficient had been negative, the same flip rule would apply to every part of the chain simultaneously, and the order of $-2$ and $3$ would swap.

Common Calculation Traps

• Forgetting the flip. The single most common error is dividing or multiplying by a negative number and leaving the symbol unchanged. If a quick test value contradicts your answer, suspect this first.
• Flipping for the wrong reason. Adding or subtracting a negative number does not flip the sign — only multiplying or dividing by a negative does. $x - 7 > 2$ becomes $x > 9$, with no flip.
• Treating the answer as one number. $x \ge 2$ means infinitely many values, not just $2$. Always read the result as a range.
• Open vs. closed endpoints. A strict symbol ($<, >$) gives an open dot and a parenthesis; an "or equal" symbol ($\le, \ge$) gives a closed dot and a bracket. Mixing these up changes the answer.
• Dividing by a variable of unknown sign. Never divide both sides by an expression like $x$ unless you know whether it is positive or negative, because the direction would depend on that sign. Rearrange to compare against zero instead.

When you need a worked solution with the number line drawn for you, paste the inequality into the GPAI Solver and it isolates the variable, flags any sign flip, and shades the interval step by step — a fast way to check your own work and build the instinct for when the symbol turns.