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# Multi-Step Inequalities

## Solve inequalities with fractions and distribution

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Solving Multi-Step Inequalities

Adding antifreeze to a car's cooling system can lower the freezing point of the water-based liquids that enable the vehicle to run. (It also raises the boiling point of these liquids.) You add enough antifreeze to lower the freezing point of the liquids to at most \begin{align*}-35^\circ C\end{align*} (in Celsius). Write the described inequality in Fahrenheit.

### Multi-Step Inequalities

Like multi-step equations, multi-step inequalities can involve having variables on both sides, the Distributive Property, and combining like terms. Again, the only difference when solving inequalities is the sign must be flipped when multiplying or dividing by a negative number.

Let's check whether \begin{align*}x = -3\end{align*} is a solution to \begin{align*}2(3x-5) \le x+10\end{align*}.

Plug in -3 for \begin{align*}x\end{align*} and see if the inequality is true.

\begin{align*}2(3(-3)-5) & \le (-3)+10\\ 2(-9-5) & \le 7\\ 2 \cdot -14 & \le 7\\ -28 & \le 7\end{align*}

This is a true inequality statement. -3 is a solution.

Now, let's solve the following multi-step inequalities.

1. Solve and graph the inequality from the problem above.

First, distribute the 2 on the left side of the inequality.

\begin{align*}2(3x-5) & \le x+10\\ 6x-10 & \le x+10\end{align*}

Now, subtract the \begin{align*}x\end{align*} on the right side to move it to the left side of the inequality. You can also add the 10’s together and solve.

\begin{align*}& \ 6x-\bcancel{10} \ge \bcancel{x}+10\\ & \underline{-x+\bcancel{10} \ - \bcancel{x}+10}\\ & \qquad \frac{\bcancel{5}x}{\bcancel{5}} \le \frac{20}{5}\\ & \qquad \ \ x \le 4\end{align*}

Test a solution, \begin{align*}x = 0: 2(3(0)-5) \le 0+10 \checkmark\end{align*}

\begin{align*}-10 \le 10\end{align*}

The graph looks like:

1. Solve \begin{align*}8x-5-4x \ge 37-2x\end{align*}.

First, combine like terms on the left side. Then, solve for \begin{align*}x\end{align*}.

\begin{align*}& 8x-5-4x \ge 37-2x\\ & \qquad 4x-\bcancel{5} \ge 37-\bcancel{2x}\\ & \underline{\quad +2x + \bcancel{5} \ +5+\bcancel{2x}}\\ & \qquad \quad \ \ \frac{\bcancel{6}x}{\bcancel{6}} \ge \frac{42}{6}\\ & \qquad \quad \quad \ x \ge 7\end{align*}

Test a solution, \begin{align*}x = 10: 8(10)-5-4(10) \ge 37-2(10)\end{align*}

\begin{align*}80-5-40 \ge 37-20 \end{align*}

\begin{align*}35 \ge 17 \checkmark\end{align*}

### Examples

#### Example 1

Earlier, you were asked to write the inequality regarding antifreeze in Fahrenheit.

Recall that the conversion formula for Celsius to Fahrenheit is \begin{align*}C= \frac{5}{9}(F-32)\end{align*}. The temperature can be equal to or greater than \begin{align*}-35^\circ C\end{align*}.

\begin{align*}\frac{5}{9}(F-32) \ge -35 \\ F-32 \ge -35 \cdot {9}{5} \\ F-32 \ge -63 \\ F \ge -31 \end{align*}

So, the temperature can be equal to or higher than \begin{align*}-31^\circ F\end{align*}.

#### Example 2

Is \begin{align*}x = 12\end{align*} a solution to \begin{align*}-3(x-10)+18 \ge x-25\end{align*}?

Plug in 12 for \begin{align*}x\end{align*} and simplify.

\begin{align*}-3(12-10)+18 &\ge 12-35\\ -3 \cdot 2 + 18 & \ge -13\\ -6+18 & \ge -13\end{align*}

This is true because \begin{align*}12 \ge -13\end{align*}, so 12 is a solution.

#### Example 3

Solve and graph \begin{align*}-(x+16)+3x>8\end{align*}.

Distribute the negative sign on the left side and combine like terms.

\begin{align*}& -(x+16)+3x>8\\ & \quad -x-16+3x>8\\ & \qquad \quad \ \ 2x-\bcancel{16}>8\\ & \underline{\qquad \qquad \quad +\bcancel{16} +16 \; \; \;}\\ & \qquad \qquad \quad \ \ \frac{\bcancel{2}x}{\bcancel{2}}>\frac{24}{2}\\ & \qquad \qquad \qquad \ x>12\end{align*}

Test a solution, \begin{align*}x = 15:\end{align*}

\begin{align*}-(15+16)+3(15) &> 8\\ -31+45 &> 8 \\ 14 &> 8\end{align*}

#### Example 4

Solve and graph \begin{align*}24-9x<6x-21\end{align*}.

First, add \begin{align*}9x\end{align*} to both sides and add 21 to both sides.

\begin{align*}& \ 24-\bcancel{9x}<6x-21\\ & \underline{\quad \ + \bcancel{9x} \ + 9x \; \; \; \; \; \; \; \; \; \;}\\ & \qquad \ 24<15x-\bcancel{21}\\ & \underline{\quad \ \ +21 \qquad \ +\bcancel{21} \; \;}\\ & \qquad \frac{45}{15} < \frac{\bcancel{15}x}{\bcancel{15}}\\ & \qquad \ \ 3

Test a solution, \begin{align*}x = 10:\end{align*}

\begin{align*}24-9(10) &< 6(10)-21\\ 24-90 &< 60-21 \\ -66 &< 39\end{align*}

### Review

Determine if the following numbers are solutions to \begin{align*}-7(2x-5)+12>-4x-13\end{align*}.

1. \begin{align*}x=4\end{align*}
2. \begin{align*}x=10\end{align*}
3. \begin{align*}x=6\end{align*}

Solve and graph the following inequalities.

1. \begin{align*}2(x-5) \ge 16\end{align*}
2. \begin{align*}-4(3x+7)<20\end{align*}
3. \begin{align*}15x-23>6x-17\end{align*}
4. \begin{align*}5x+16+2x\le - 19\end{align*}
5. \begin{align*}4(2x-1)\ge3(2x+1)\end{align*}
6. \begin{align*}11x-17-2x \le - (x-23)\end{align*}

Solve the following inequalities.

1. \begin{align*}5-5x>4(3-x)\end{align*}
2. \begin{align*}-(x-1)+10<-3(x-3)\end{align*}
3. Solve \begin{align*}5x+4 \le -2(x+3)\end{align*} by adding the \begin{align*}2x\end{align*} term on the right to the left-hand side.
4. Solve \begin{align*}5x+4 \le-2(x+3)\end{align*} by subtracting the \begin{align*}5x\end{align*} term on the left to the right-hand side.
5. Compare your answers from 12 and 13. What do you notice?
6. Challenge Solve \begin{align*}3x-7>3(x+3)\end{align*}. What happens?

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 1.11.

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### Vocabulary Language: English

linear equation

A linear equation is an equation between two variables that produces a straight line when graphed.

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