2.05 Multi-step inequalities
Introduction
We were introduced to writing, solving, and representing inequalities on number lines in 6th grade. In 7th grade, we may have seen and worked with modeling real-world scenarios with inequalities. Now, we'll continue to reason with inequalities and apply the properties of inequality.
Multi-step inequalities
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.
| \text{Asymmetric property of inequality} | \text{If } a>b, \text{then } b<a |
| \text{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.
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.
Exploration
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 < 4 | \text{Add } 2 \text{ to both sides} | ||
| 6 > -2 | \text{Subtract } 2 \text{ from both sides} | ||
| 3 < 10 | \text{Multiply by } 2 \text{ on both sides} | ||
| 1 > -7 | \text{Multiply by } -2 \text{ on both sides} | ||
| 4 > 2 | \text{Divide by } 2 \text{ on both sides} | ||
| -8 < 12 | \text{Divide by } -2 \text{ on both sides} |
1. Did any operations cause the given inequality to become a false inequality?
2. 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:
| \text{Addition property of inequality} | \text{If } a>b, \text{then } a+c>b+c |
| \text{Subtraction property of inequality} | \text{If } a>b, \text{then } a-c>b-c |
| \text{Multiplication property of inequality} | \text{If } a>b \text{ and } c>0, \text{then } ac>bc \text{ or if } a>b \text{ and } c<0, \text{then } ac<bc |
| \text{Division property of inequality} | \text{If } a>b \text{ and } c>0, \text{then } \dfrac{a}{c}>\dfrac{b}{c} \text{ or if } a>b \text{ and } c<0, \text{then } \dfrac{a}{c}<\dfrac{b}{c} |
Examples
Example 1
Consider the inequality \dfrac{-8-3x}{2} \leq 5
Solve the inequality
Plot the inequality on a number line.
Is x=3 a viable or nonviable solution to the inequality?
Example 2
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.
Write an inequality that represents the number of foils Pauline could get.
How many foils could Pauline get and still afford the styling?
Determine whether N=-2 is a viable solution to the inequality in the context of the question.
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>b and c<0, then a \cdot c < b \cdot c
- If a>b and c<0, then a \div c < b \div c
Because inequalities have infinite solutions, inequalities used to represent real-world situations often include solutions that are unreasonable in context and therefore non-viable.