Suppose that you know that 10 less than 3 times the number of coins in your piggy bank is greater than 200. If the number of coins in your piggy bank is represented by , how would you go about finding the value of this variable?

### Multi-Step Inequalities

Previously we worked on one-step inequalities. Inequalities, like equations, can require several steps to isolate the variable. These inequalities are called **multi-step inequalities.** With the exception of the Multiplication/Division Property of Inequality, the process of solving multi-step inequalities is identical to solving multi-step equations.

**Procedure to Solve an Inequality:**

**Step 1: **Remove any parentheses by using the Distributive Property.

**Step 2: **Simplify each side of the inequality by combining like terms.

**Step 3: **Isolate the term. Use the Addition/Subtraction Property of Inequality to get the variable on one side of the inequality sign and the numerical values on the other.

**Step 4: **Isolate the variable. Use the Multiplication/Division Property of Inequality to get the variable alone on one side of the inequality. Remember to reverse the inequality sign if you are multiplying or dividing by a negative number.

**Step 5: **Check your solution.

#### Let's solve the following inequalities:

- Solve for .

Begin by using the checklist above.

**Step 1:** Parentheses? No

**Step 2:** Like terms on the same side of inequality? No

**Step 3:** Isolate the term using the Addition Property.

Simplify.

**Step 4:** Isolate the variable using the Multiplication or Division Property.

**Step 5:** Check your solution. Choose a number less than 2.5, say 0, and check using the original inequality.

Yes, the answer checks.

- Solve for .

Begin by using the checklist above.

**Step 1:** Parentheses? No

**Step 2:** Like terms on the same side of inequality? No

**Step 3:** Isolate the term using the Addition Property.

Simplify.

**Step 4:** Isolate the variable using the Multiplication or Division Property.

Because the number you are dividing by is negative, you must reverse the inequality sign.

**Step 5:** Check your solution by choosing a number larger than 3.75, such as 10.

#### Identifying the Number of Solutions to an Inequality

Inequalities can have infinitely many solutions, no solutions, or a finite set of solutions. Most of the inequalities you have solved to this point have an infinite number of solutions. By solving inequalities and using the context of a problem, you can determine the number of solutions an inequality may have.

#### Let's solve the following problems and identify how many solutions they have:

Begin by isolating the variable using the Addition Property of Inequality.

Simplify.

This is an untrue inequality. Negative five is never greater than six. Therefore, the inequality has no solutions.

- Suppose you were given the statement “The speed limit is 65 miles per hour.” Use inequalities and interval notation to describe the set of possible speeds at which a car could drive under the speed limit.

The speed at which you drive cannot be negative, which means , and it must be less than 65 miles per hour, so . Combining these we get . Therefore, the set of possibilities using interval notation is [0, 65].

This solution set has infinitely many solutions, since there are infinitely many real numbers between 0 and 65.

### Examples

#### Example 1

Earlier, you were told that 10 less than 3 times the number of coins in your piggy bank is greater than 200. If the number of coins in your piggy bank is represented by , what is the value of this variable?

First, you need to write an inequality that represents this situation:

Now, follow the checklist.

**Step 1:** Parentheses? No.

**Step 2:** Like terms on the same side of inequality? No.

**Step 3:** Isolate the term using the Addition Property.

**Step 4:** Isolate the variable using the Multiplication or Division Property.

**Step 5:** Check your solution by choosing a number less than 70 such as 60.

You have less than 70 coins in your piggy bank.

#### Example 2

Solve for .

Begin by using the checklist.

**Step 1:** Parentheses? Yes. Use the Distributive Property to clear the parentheses.

Simplify.

**Step 2:** Like terms on the same side of inequality? Yes. Combine these.

**Step 3:** Isolate the term using the Addition Property.

Simplify.

**Step 4:** Isolate the variable using the Multiplication or Division Property.

**Step 5:** Check your solution by choosing a number less than 3, such as –5.

### Review

In 1–15, solve each of the inequalities and graph the solution set.

**Mixed Review**

- Solve:
- Graph on a coordinate plane.
- Classify using the real number hierarchy.
- What are some problem-solving methods you have learned so far in this textbook? List one example for each method.
- A circle has an area of . What is the radius of a circle with area of ?
- Solve for

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.4.