Some mathematical relations compare two non-equivalent expressions. These are known as **inequalities**.

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

Asymmetric property of inequality | \text{If } a>b, \text{then } b<a |
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Transitive property of inequality | \text{If } a>b \text{ and } b>c, \text{then } a>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.

Different notations can be used to represent algebraic solutions, such as inequalities, set notation, and interval notation.

**Set notation** uses the notation \left\{â¬š \middle\vert â¬š \right\}, where the vertical line is read as "such that" and defines the variable used. **Interval notation** uses parentheses \left( \enspace \right) to indicate values that are never reached and not included in the solution set. Square brackets indicate\left[ \enspace \right] closed intervals that include the endpoints in a solution set.

Based on the context, some values might be calculated algebraically, but are not reasonable based on the restrictions of the scenario. For example, time and lengths generally cannot be negative, which can create restrictions on the possible values x and y can take on.

Complete the following chart by performing the indicated operations:

Consider the inequality | Perform the operation on the inequality | Write the new inequality | True or false? |
---|---|---|---|

1 \lt 4 | \text{Add } 2 \text{ to both sides} | ||

6 \gt -2 | \text{Subtract } 2 \text{ from both sides} | ||

3 \lt 10 | \text{Multiply by } 2 \text{ on both sides} | ||

1 \gt -7 | \text{Multiply by } -2 \text{ on both sides} | ||

4 \gt 2 | \text{Divide by } 2 \text{ on both sides} | ||

-8 \lt 12 | \text{Divide by } -2 \text{ on both sides} |

Did any operations cause the given inequality to become a false inequality?

Can you think of something to change about a false inequality without changing the operation performed?

When multiplying or dividing an inequality by a negative value the inequality symbol is reversed.

The properties of inequality are:

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

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

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

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

a

Solve the inequality

Worked Solution

b

Plot the inequality on a number line.

Worked Solution

c

Is x=3 a viable or nonviable solution to the inequality?

Worked Solution

Calandra charges \$ 37.72 to style hair, as well as an additional \$ 6 per foil. Pauline would like the total cost for her styling to be no more than \$ 95.86.

a

Write an inequality that represents the number of foils Pauline could get.

Worked Solution

b

How many foils could Pauline get and still afford the styling?

Worked Solution

c

Determine whether N=-2 is a viable solution to the inequality in the context of the question.

Worked Solution

Solve the inequality 4\left(x+5\right) \lt 3\left(2-x\right).

Worked Solution

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:

If a\gt b and c\gt 0, then a \cdot c \lt b \cdot c

If a \gt b and c \lt 0, then \dfrac{a}{c} \lt \dfrac{b}{c}

The solution set of an inequality is the set of values that makes the inequality true.

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