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 \begin{align*}c\end{align*}, 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 \begin{align*}ax\end{align*} 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{align*}w: \ 6x-5<10\end{align*}.

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 \begin{align*}ax\end{align*} term using the Addition Property.

\begin{align*}6x-5+5<10+5\end{align*}

Simplify.

\begin{align*}6x<15\end{align*}

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

\begin{align*}\frac{6x}{6} < \frac{15}{6} = x < \frac{5}{2}\end{align*}

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

\begin{align*}6(0)-5 &< 10?\\ -5 & < 10\end{align*}

Yes, the answer checks. \begin{align*}x < 2.5\end{align*}

- Solve for \begin{align*}x: \ -9x<-5x-15\end{align*}.

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 \begin{align*}ax\end{align*} term using the Addition Property.

\begin{align*}-9x+5x<-5x+5x-15\end{align*}

Simplify.

\begin{align*}-4x<-15\end{align*}

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

\begin{align*}\frac{-4x}{-4} < \frac{-15}{-4}\end{align*}

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

\begin{align*}x>\frac{15}{4} \rightarrow x > 3 \frac{3}{4}\end{align*}

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

\begin{align*}-9(10)& <-5(10)-15?\\ \checkmark \ -90 & <-65\end{align*}

#### 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{align*}x-5>x+6\end{align*}

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

\begin{align*}x-x-5>x-x+6\end{align*}

Simplify.

\begin{align*}-5>6\end{align*}

This is an untrue inequality. Negative five is never greater than six. Therefore, the inequality \begin{align*}x-5>x+6\end{align*} 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 \begin{align*}0\le s\end{align*}, and it must be less than 65 miles per hour, so \begin{align*}s \le 65.\end{align*}. Combining these we get \begin{align*}0\le s \le 65\end{align*}. 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 \begin{align*}c\end{align*}, what is the value of this variable?

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

\begin{align*}3c-10>200\end{align*}

Now, follow the checklist.

**Step 1:** Parentheses? No.

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

**Step 3:** Isolate the \begin{align*}ax\end{align*} term using the Addition Property.

\begin{align*}3c - 10 + 10 &< 200 + 10\\ 3c& < 210\end{align*}

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

\begin{align*}3c\div 3 &< 210 \div 3\\ c &< 70\end{align*}

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

\begin{align*}3(60) - 10 &< 200\\ 180-10 &< 200\\ \checkmark 170 &<200\end{align*}

You have less than 70 coins in your piggy bank.

#### Example 2

Solve for \begin{align*}x: \ 4x-2(3x-9) \le -4(2x-9)\end{align*}.

Begin by using the checklist.

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

\begin{align*}4x+(-2)(3x)+(-2)(-9)\le-4(2x)+(-4)(-9)\end{align*}

Simplify.

\begin{align*}4x-6x+18 \le -8x+36\end{align*}

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

\begin{align*}-2x+18\le-8x+36\end{align*}

**Step 3:** Isolate the \begin{align*}ax\end{align*} term using the Addition Property.

\begin{align*}-2x+8x+18\le-8x+8x+36\end{align*}

Simplify.

\begin{align*}6x+18 & \le36\\ 6x+18-18 & \le36-18\\ 6x & \le18\end{align*}

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

\begin{align*}\frac{6x}{6} \le \frac{18}{6} \rightarrow x \le 3\end{align*}

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

\begin{align*}4(-5)-2(3 \cdot -5-9) & \le -4(2 \cdot -5-9)\\ \checkmark \ 28 & <76\end{align*}

### Review

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

- \begin{align*}6x-5 < 10\end{align*}
- \begin{align*}-9x < -5x-15\end{align*}
- \begin{align*}-\frac{9x}{5} \le 24\end{align*}
- \begin{align*}\frac{9x}{5}-7 \ge -3x+12\end{align*}
- \begin{align*}\frac{5x-1}{4} > -2 (x+5)\end{align*}
- \begin{align*}4x+3 < -1\end{align*}
- \begin{align*}2x < 7x - 36\end{align*}
- \begin{align*}5x >8x + 27\end{align*}
- \begin{align*}5 - x < 9 + x\end{align*}
- \begin{align*}4-6x \le 2(2x+3)\end{align*}
- \begin{align*}5(4x+3)\ge 9(x-2)-x\end{align*}
- \begin{align*}2(2x-1)+3 < 5(x+3)-2x\end{align*}
- \begin{align*}8x-5(4x+1) \ge -1+2(4x-3)\end{align*}
- \begin{align*}2(7x-2)-3(x+2)< 4x-(3x+4)\end{align*}
- \begin{align*}\frac{2}{3}x-\frac{1}{2}(4x-1) \ge x+2(x-3)\end{align*}

**Mixed Review**

- Solve: \begin{align*}10 \ge -5f.\end{align*}
- Graph \begin{align*}y=-7\end{align*} on a coordinate plane.
- Classify \begin{align*}\sqrt{5}\end{align*} 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 \begin{align*}A=\pi r^2\end{align*}. What is the radius of a circle with area of \begin{align*}196\pi \ in^2\end{align*}?
- Solve for \begin{align*}a: \frac{6}{a}=\frac{-22}{a+4}.\end{align*}

### Review (Answers)

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