Linear Inequalities — Definition, Rules, and Examples

What Are the Rules for Solving Linear Inequalities?

You solve a linear inequality almost exactly like a linear equation, isolate the variable, with one rule that has no equation equivalent.

• Add or subtract anything, both sides, the sign stays. $x + 5 \leq 9 \implies x \leq 4$.

• Multiply or divide by a positive number, the sign stays. $3x < 12 \implies x < 4$.

• Multiply or divide by a negative number, the sign flips. This is the rule. $-2x < 6 \implies x > -3$ (the $<$ became $>$).

The flip is not arbitrary. Multiplying by a negative reverses order on the number line: $2 < 3$ is true, but multiply both sides by $-1$ and $-2 < -3$ is false, you have to write $-2 > -3$ to keep it true. So whenever a negative factor crosses the inequality, the symbol reverses.

Why does the inequality sign flip when you divide by a negative?

Because "bigger" and "smaller" swap places under negation. Picture $4$ and $7$ on the number line, where $4 < 7$. Their negatives $-4$ and $-7$ sit on the mirror-image side, and now $-4 > -7$ (the larger original became the smaller negative).

Dividing or multiplying an inequality by a negative does exactly this mirroring, so the symbol must turn around to stay true. Forgetting this single flip is the number-one error in the topic.

Types of Linear Inequalities

• One variable: $3x - 1 > 5$. The solution is an interval on the number line.

• Two variables: $2x + y \leq 6$. The solution is a half-plane on the coordinate grid, every point on one side of the boundary line.

A single linear inequality is not the same as a compound inequality, which joins two inequalities with "and"/"or" (like $1 < x \leq 5$). This article stays on the single inequality; the compound case has its own logic and its own page.

Examples of Linear Inequalities

Example 1

Solve $x + 7 < 12$.

Subtract $7$ from both sides:

$x + 7 - 7 < 12 - 7$

$x < 5$.

Final answer: $x < 5$, every value below $5$.

Example 2

Solve $-3x \geq 9$.

A student divides both sides by $-3$ and writes $x \geq -3$.

Here is the wrong path, and it is the most instructive one in this topic. Dividing by $-3$ as if it were a positive leaves the $\geq$ untouched.

Test it. If $x \geq -3$ were right, then $x = 0$ should work: $-3(0) = 0 \geq 9$? That is false. The "solution" includes values that break the original inequality, so it cannot be right.

The rescue: dividing by a negative flips the sign.

$-3x \geq 9$

$x \leq -3$ (divide by $-3$, flip $\geq$ to $\leq$).

Test $x = -4$: $-3(-4) = 12 \geq 9$. ✓

Final answer: $x \leq -3$.

Example 3

Solve $2x + 3 \leq 11$ and graph it.

Subtract $3$:

$2x \leq 8$.

Divide by $2$ (positive, sign stays):

$x \leq 4$.

On the number line: a closed circle at $4$ (because $\leq$ includes $4$), shaded left.

Final answer: $x \leq 4$.

Example 4

Solve $5 - 2x > 1$.

Subtract $5$:

$-2x > -4$.

Divide by $-2$ (negative, flip $>$ to $<$):

$x < 2$.

Final answer: $x < 2$. In interval notation that is $(-\infty, 2)$.

Example 5

Solve $\dfrac{x}{3} - 2 \geq 4$.

Add $2$:

$\dfrac{x}{3} \geq 6$.

Multiply by $3$ (positive, sign stays):

$x \geq 18$.

Final answer: $x \geq 18$, or $[18, \infty)$ in interval notation.

Example 6

Solve the two-variable inequality $x + y \leq 4$ and describe its graph.

First treat the boundary as an equation: $x + y = 4$ is the line through $(4, 0)$ and $(0, 4)$.

Because the relation is $\leq$ (includes equality), the boundary line is solid.

Test a point not on the line, the origin $(0,0)$: $0 + 0 = 0 \leq 4$? True.

So shade the side of the line containing the origin.

Final answer: the solution is the closed half-plane on the origin's side of the solid line $x + y = 4$.

Why Inequalities Run the Real World

"Why not just use equations?" Because most decisions are about staying within a limit, not hitting it exactly, and a limit is an inequality.

• Constraints are inequalities by nature. A budget says spending $\leq$ income. A speed limit says $v \leq 60$. A safety factor says load capacity $\geq$ expected load. None of these is an equals sign.

• Optimisation lives on inequalities. Linear programming, the method that schedules airlines, routes deliveries, and plans factory output, is built entirely on systems of linear inequalities, finding the best point inside a region they fence off.

• The boundary is where the danger is. Treating $\geq$ as $=$ ignores the safety margin. The 1940 Tacoma Narrows Bridge collapse is the textbook reminder that a structure rated for "up to" a load behaves very differently once the inequality is violated, the margin is the point, not the rounding error.

The destination, then, is not "solve for $x$." It is: describe the entire safe region, and know which boundary you must not cross.

Where Linear Inequalities Trip Students Up

Mistake 1: Forgetting to flip the sign with a negative

Where it slips in: any step that divides or multiplies by a negative number.

Don't do this: treat $-2x > 6$ like an equation and write $x > -3$. The memorizer who learned "do the same to both sides" applies it without the negative-number exception.

The correct way: the moment a negative factor crosses the inequality, reverse the symbol: $-2x > 6 \implies x < -3$. A reliable habit is to test one value from your answer back in the original, it catches a missed flip instantly.

Mistake 2: Using the wrong circle when graphing

Where it slips in: drawing the solution on a number line.

Don't do this: the rusher uses a filled dot for $<$ or an open dot for $\leq$ without thinking about which boundary is included.

The correct way: open circle for strict $<$ or $>$ (boundary excluded); closed circle for $\leq$ or $\geq$ (boundary included). The symbol's "or equal to" half is exactly what fills the dot in.

Mistake 3: Shading the wrong half-plane in two variables

Where it slips in: graphing two-variable inequalities.

Don't do this: the second-guesser draws the boundary line correctly, then shades whichever side "looks right."

The correct way: pick a test point not on the line, the origin $(0,0)$ is easiest when the line misses it, substitute it into the inequality, and shade the side it lands on if the statement is true. Never guess the side; test it.

Conclusion

• A linear inequality compares two first-degree expressions with $<$, $>$, $\leq$, or $\geq$, and its solution is a range of values.

• You solve it like an equation, with one exception: multiplying or dividing by a negative flips the sign.

• On a number line, use an open circle for $<$/$>$ and a closed circle for $\leq$/$\geq$.

• A two-variable linear inequality graphs as a half-plane, with the correct side found by a test point.

• A single linear inequality differs from a compound inequality, which joins two with "and" or "or."

A Practical Next Step

Practice these to solidify your understanding:

1. Solve $4 - x \leq 7$ and graph it.

2. Solve $-5x + 2 > 17$ and state the answer in interval notation.

3. Graph $2x - y > 2$ in the coordinate plane using a test point.

If the sign-flip rule still feels uncertain, redo Example 2 and verify your answer by substitution. To master linear inequalities with a teacher, explore Bhanzu's algebra tutor, algebra classes, or math tutoring. Want a live Bhanzu trainer to walk through inequalities with your child? Book a free demo class.

Read More

• Linear Equations: Solving for a single value before moving to inequalities
  Absolute Value: The step before absolute-value inequalities
  Increasing and Decreasing Intervals: reading solution ranges off a graph