# Writing and Graphing One-Variable Inequalities

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Equations are mathematical statements relating two equal quantities. There are also times when it's necessary to relate two quantities that are

**not**equal. For this,*inequalities*are used.## Inequality

An **inequality**, similar to an equation, is a mathematical statement *comparing* two quantities. Some inequalities indicate that the two quantities are *not necessarily* equal, while others are *strictly never* equal.

Symbol | Meaning | Strict / Non-strict |
---|---|---|

$\lt$ | is less than | Strict |

$\leq$ | is less than or equal to | Non-strict |

$\gt$ | is greater than | Strict |

$\geq$ | is greater than or equal to | Non-strict |

## Solution Set of an Inequality

When an inequality contains an unknown variable, it's possible to solve the inequality. A solution of an inequality is any value of the variable that makes the inequality true. For instance, the inequality $x + 2 < 5$

has the solution $x = 1,$ because replacing $x$ with $1$ yields $3 < 5,$ a true statement. Notice that $x=1$ is not the only value that solves $x+2<5;$ $x=0$ and $x=2$ also work. In fact, most inequalities have an infinite number of solutions. The set of these solutions is called the*solution set.*

## Graphing an Inequality on a Number Line

The graph of a one-variable inequality is a visual representation of the inequality's solution set, which can be drawn on a number line in three steps:

- Identify if the inequality is strict.
- Draw the boundary point;
- use an open point $(\circ)$ for strict or
- use a closed point $(\bullet)$ for non-strict.
- Shade the rest of the solution set.

An arrow in either direction indicates that all numbers in that direction are part of the solution set.

$x \ge 0$

$x \leq 2$

$x > \text{-} 3$

## Exercises

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