We can solve inequalities by using various properties to isolate the variable, in a similar way to solving equations.

The properties of inequality are:

\text{Addition property of inequality} | \text{If } a \lt b, \text{then } a+c \lt b+c; \\\ \text{If } a \gt b, \text{then } a+c \gt b+c |

\text{Subtraction property of inequality} | \text{If } a\lt b, \text{then } a-c \lt b-c; \\\ \text{If } a \gt b, \text{then } a-c \gt b-c |

\text{Multiplication property of inequality} | \text{If } a \lt b \text{ and } c \gt 0, \text{then } a \cdot c \lt b\cdot c; \\\ \text{if } a\gt b \text{ and } c \gt 0, \text{then } a\cdot c \gt b\cdot c |

\text{Division property of inequality} | \text{If } a\lt b \text{ and } c \gt 0, \text{then } \dfrac{a}{c} \lt \dfrac{b}{c};\\\ \text{ if } a \gt b \text{ and } c \gt 0, \text{then } \dfrac{a}{c}\gt\dfrac{b}{c} |

\text{Asymmetric property of inequality} | \text{If } a\gt b, \text{then } b \lt a |

\text{Transitive property of inequality} | \text{If } a \gt b \text{ and } b \gt c, \text{then } a \gt c |

Solving an inequality using the properties of inequalities results in a **solution set**.

We can represent solutions to inequalities **algebraically**, by using numbers, letters, and/or symbols, or **graphically**, by using a coordinate plane or number line.

Remember: when multiplying or dividing an inequality by a negative value the inequality symbol is reversed.

Consider the inequality \dfrac{-8-3x}{2} \leq 5

a

Solve the inequality

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b

Plot the inequality on a number line.

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c

Is x=3 in the solution set for the inequality?

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Arlene charges \$ 42.75 for a pet grooming session, plus an additional \$ 5 for each special treatment. Clarisse wants the total cost for her pet's grooming to be no more than \$ 102.80.

a

Write an inequality that represents the number of special treatments Clarisse could get for her pet.

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b

How many special treatments could Clarisse get for her pet and still afford the pet grooming?

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c

Is s=-2 a viable solution?

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Idea summary

Just like the properties of equality, the properties of inequality can justify how we solve inequalities.

The multiplication and division properties of inequality change the meaning of an inequality when multiplying or dividing by a negative number, meaning we have to reverse the inequality symbol when applying the property.

Because inequalities have infinitely many solutions, inequalities used to represent real-world situations often include solutions that are unreasonable in context and therefore non-viable.