### Addition Property of Inequality

In a **linear inequality in one variable**, each term can be written as a number or a product of the variable and a number. The variable is not raised to a power, in the denominator of a fraction, or under a radical such as a square root. A linear inequality is the same as a linear equation except it has an inequality symbol ($\lt$, $\leq$, $\gt$, or $\geq$) instead of an equal sign.

A **solution of an inequality** is a value of the variable that makes the inequality true, which means the comparison is valid. The process for solving linear inequalities is similar to the process for linear equations. Properties of inequality are used to isolate the variable. The difference is there can be infinitely many solutions of an inequality.

The **addition property of inequality** states that the solution of an inequality does not change after adding the same number to both sides of the inequality.

Addition Property of Inequality | Examples |
---|---|

If the same quantity is added to both sides of a true inequality, the resulting inequality is still true. |
If $a \lt b$, then:
$a+c \lt b+c$ $a+c\leq b+c$ $a+c \gt b+c$ $a+c\geq b+c$ |

The solutions of the inequality are all numbers less than 10. This can be expressed in different ways.

- Inequality: $x \lt 10$
- Interval notation: $(-\infty,10)$
- Set-builder notation: $\{x|x \lt 10\}$

### Subtraction Property of Inequality

**subtraction property of inequality**states that the solution of an inequality does not change after subtracting the same number from both sides of the inequality. This property is analogous to the subtraction property of equality.

Subtraction Property of Inequality | Examples |
---|---|

If the same quantity is subtracted from both sides of a true inequality, the resulting inequality is still true. |
If $a \lt b$, then:
$a-c\lt b-c$ $a-c\leq b-c$ $a-c\gt b-c$ $a-c\geq b-c$ |

The solutions of the inequality are all numbers greater than 4. This can be expressed in different ways.

- Inequality: $x \gt 4$
- Interval notation: $(4,\infty)$
- Set-builder notation: $\{x|x \gt 4\}$

### Multiplication Property of Inequality

The **multiplication property of inequality** states that the solution of an inequality does not change if:

- both sides of the inequality are multiplied by the same positive number, or
- both sides of the inequality are multiplied by the same negative number and the inequality sign is reversed

Notice that the positive value is greater than zero and the negative value is less than zero. Keeping the inequality sign the same would produce a false statement because $-3\ngtr0$. This is why the inequality symbol is reversed.

Multiplication Property of Inequality | Example |
---|---|

If both sides of a true inequality are multiplied by the same positive quantity, the resulting inequality is still true. |
If $a\lt b$ and $c\gt 0$, then:
$ac\lt bc$ $ac\leq bc$ $ac\gt bc$ $ac\geq bc$ |

If both sides of a true inequality are multiplied by the same negative quantity and the inequality symbol is reversed, the resulting inequality is still true. |
If $a\lt b$ and $c\lt 0$, then:
$ac\gt bc$ $ac\geq bc$ $ac\lt bc$ $ac\leq bc$ |

The solutions of the inequality are all numbers less than or equal to 27. This can be expressed in different ways:

- Inequality: $x \leq 27$
- Interval notation: $(-\infty,27]$
- Set-builder notation: $\{x|x\leq27\}$

### Division Property of Inequality

Like the multiplication property of inequality, the division property of inequality has two parts.

The **division property of inequality** states that the solution of an inequality does not change if:

- both sides of the inequality are divided by the same positive number, or
- both sides of the inequality are multiplied by the same negative number and the inequality sign is reversed

Division Property of Inequality | Example |
---|---|

If both sides of a true inequality are divided by the same positive quantity, the resulting inequality is still true. |
If $a\lt b$ and $c\gt 0$, then:
$\frac{a}{c}\lt \frac{b}{c}$ If $a\leq b$ and $c\gt 0$, then: $\frac{a}{c}\leq \frac{b}{c}$ If $a>b$ and $c\gt 0$, then: $\frac{a}{c}\gt \frac{b}{c}$ If $a\geq b$ and $c\gt 0$, then: $\frac{a}{c}\geq \frac{b}{c}$ |

If both sides of a true inequality are divided by the same negative quantity and the inequality symbol is reversed, the resulting inequality is still true. |
If $a\lt b$ and $c\lt 0$, then:
$\frac{a}{c}\gt \frac{b}{c}$ If $a\leq b$ and $c\lt 0$, then: $\frac{a}{c}\geq \frac{b}{c}$ If $a>b$ and $c\lt 0$, then: $\frac{a}{c}\lt \frac{b}{c}$ If $a\geq b$ and $c\lt 0$, then: $\frac{a}{c}\leq \frac{b}{c}$ |

The solutions of the inequality are all numbers greater than or equal to 3. This can be expressed in different ways:

- Inequality: $x \geq 3$
- Interval notation: $[3,\infty)$
- Set-builder notation: $\{x|x\geq3\}$