# 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](/content/math/algebra/compound-inequality/index.html), 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:

1. Solve 4−x≤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.