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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.

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 |

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 theIs $r = 2$ an element of the solution set of the following inequality? $\dfrac{3}{r} \leq \text{-} 1$

Show Solution

If a value is a solution to an inequality, it is an element of the solution set. To determine if $r=2$ is a solution, we can substitute the value into the inequality and evaluate. If the resulting inequality is true, the value is a solution; if it's false, it is not a solution.

$\dfrac{3}{r} \stackrel{?}{\leq} \text{-} 1$

Substitute$r={\color{#0000FF}{2}}$

$\dfrac{3}{{\color{#0000FF}{2}}} \stackrel{?}{\leq} \text{-} 1$

WriteDecWrite as a decimal

$1.5 \nleq \text{-} 1$

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.

$x \ge 0$

$x \leq 2$

$x > \text{-} 3$

Graph the inequality $x < 9$ on a number line.

Show Solution

The inequality reads $``x$ is less than $9,"$ which means that $x = 9$ is **not** a solution to the inequality, but every value of $x$ less than $9$ is. Thus, our graph must have an open circle at $x = 9.$

Every value of $x$ less than $9$ also has to be included in the graph. Since smaller numbers lie to the left on the number line, this is graphed as an arrow pointing to the left.

We have now fully graphed the inequality $x < 9.$

Write the inequality shown by the graph.

Show Solution

To begin, let's use the variable $x$ in the inequality. From the graph, we can see a closed circle at $\text{-} 3.$ Therefore, $x = \text{-} 3$ is a solution of the inequality. The arrow pointing toward the right indicates that all values greater than $3$ are also part of the solution set. Thus, the inequality would read $``x$ is greater than or equal to $\text{-} 3,"$ which is expressed algebraically as $x \geq \text{-} 3.$

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