Linear Inequalities — Definition, Rules, and Examples

TL;DR

A linear inequality compares two linear expressions using <<<, >>>, ≤ or ≥ instead of an equals sign, so its solution is a range of values, not a single number. This article gives the solving rules, including the one rule that flips the sign, dividing or multiplying by a negative, shows how to graph the solution on a number line, and works through one- and two-variable 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≤9⟹x≤4.
Multiply or divide by a positive number, the sign stays. 3x<12⟹x<4.
Multiply or divide by a negative number, the sign flips. −2x<6⟹x>−3.

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).

Types of Linear Inequalities

One variable: 3x−1>5. The solution is an interval on the number line.
Two variables: 2x+y≤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". 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<5.

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

Example 2

Solve −3x≥9.
A student divides both sides by −3 and writes x≥−3.

The rescue: dividing by a negative flips the sign.

x≤−3.

Final answer: x≤−3.

Example 3

Solve 2x+3≤11 and graph it.
Subtract 3:

2x≤8.
Divide by 2:

x≤4.

Final answer: x≤4.

Example 4

Solve 5−2x>15.
Subtract 5:

−2x>−4.
Divide by −2 (negative, flip > to <):

x<2.

Final answer: x<2.

Example 5

Solve ( \frac{x}{3} - 2 ≥ 4.
Add 2:

( \frac{x}{3}≥6.
Multiply by 3:

x≥18.

Final answer: x≥18.

Example 6

Solve the two-variable inequality x+y≤4 and describe its graph.
First treat the boundary as an equation: x+y=4.
Because the relation is ≤ (includes equality), the boundary line is solid.

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

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

Constraints are inequalities by nature. A budget says spending ≤ income. A speed limit says v≤60. A safety factor says load capacity ≥ expected load.
Optimisation lives on inequalities. Linear programming is built entirely on systems of linear inequalities.
The boundary is where the danger is. Treating ≥ as = ignores the safety margin.

Where Linear Inequalities Trip Students Up

Mistake 1: Forgetting to flip the sign with a negative

Mistake 2: Using the wrong circle when graphing

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

Conclusion

A linear inequality compares two first-degree expressions with <<<, >>>, ≤ or ≥, 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.
A two-variable linear inequality graphs as a half-plane.
A single linear inequality differs from a compound inequality.

A Practical Next Step

Practice these to solidify your understanding:

Solve 4−x≤7 and graph it.
Solve −5x+2>17 and state the answer in interval notation.
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.