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# Multi-Step Inequalities

## Solve inequalities with fractions and distribution

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Multi-Step Inequalities

Suppose you wanted to know if you had enough money to buy a movie ticket, but only knew that one friend bought two tickets and a popcorn, and that she spent more than another friend who purchased three tickets. Could you describe the price of popcorn compared to the price of a ticket? If you knew that the situation could be described with the inequality \begin{align*}2t+p>3t,\end{align*} could you solve the inequality for \begin{align*}p\end{align*}, the popcorn price?

### Multi-Step Inequalities

Most inequalities require several steps to arrive at the solution. As with solving equations, we must use the order of operations to find the correct solution. In addition, remember that when we multiply or divide the inequality by a negative number, the direction of the inequality sign changes.

The general procedure for solving multi-step inequalities is almost exactly like the procedure for solving multi-step equations:

1. Clear parentheses on both sides of the inequality and collect like terms.
2. Add or subtract terms so the variable is on one side and the constant is on the other side of the inequality sign.
3. Multiply and divide by whatever constants are attached to the variable. Remember to change the direction of the inequality sign if you multiply or divide by a negative number.

Solve the inequality \begin{align*}\frac{9x}{5}-7 \ge -3x + 12\end{align*} and graph the solution set.

Original problem: \begin{align*}\frac{9x}{5}-7 \ge -3x+12\end{align*}

Add \begin{align*}3x\end{align*} to both sides: \begin{align*}\frac{9x}{5} + 3x - 7 \ge -3x+3x+12\end{align*}

Simplify: \begin{align*}\frac{24x}{5}-7 \ge 12\end{align*}

Add 7 to both sides: \begin{align*}\frac{24x}{5}-7+7 \ge 12+7\end{align*}

Simplify: \begin{align*}\frac{24x}{5} \ge 19\end{align*}

Multiply 5 to both sides: \begin{align*}5 \cdot \frac{24x}{5} \ge 5 \cdot 19\end{align*}

Simplify: \begin{align*}24x \ge 95\end{align*}

Divide both sides by 24: \begin{align*}\frac{24x}{24} \ge \frac{95}{24}\end{align*}

Simplify: \begin{align*}x \ge \frac{95}{24}\end{align*} Answer

Graph:

Solve the inequality \begin{align*}-25x + 12 \le -10x - 12\end{align*} and graph the solution set.

Original problem: \begin{align*}-25x+12 \le -10x-12\end{align*}

Add \begin{align*}10x\end{align*} to both sides: \begin{align*}-25x+10x+12 \le -10x+10x-12\end{align*}

Simplify: \begin{align*}-15x+12 \le -12\end{align*}

Subtract 12: \begin{align*}-15x+12-12\le -12-12\end{align*}

Simplify: \begin{align*}-15x \le -24\end{align*}

Divide both sides by -15: \begin{align*}\frac{-15x}{-15} \ge \frac{-24}{-15}\end{align*} flip the inequality sign

Simplify: \begin{align*}x \ge \frac{8}{5}\end{align*} Answer

Graph:

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

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

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

Collect like terms: \begin{align*}-2x+18 \le -8x+36\end{align*}

Add \begin{align*}8x\end{align*} to both sides: \begin{align*}-2x+8x+18 \le -8x+8x+36\end{align*}

Simplify: \begin{align*}6x+18 \le 36\end{align*}

Subtract 18: \begin{align*}6x+18-18 \le 36-18\end{align*}

Simplify: \begin{align*}6x \le 18\end{align*}

Divide both sides by 6: \begin{align*}\frac{6x}{6} \le \frac{18}{6}\end{align*}

Simplify: \begin{align*}x \le 3\end{align*} Answer

### Example

Solve the inequality \begin{align*}\frac{5x-1}{4} > -2(x+5)\end{align*}.

Original problem: \begin{align*}\frac{5x-1}{4} > -2(x+5)\end{align*}

Simplify parenthesis: \begin{align*}\frac{5x-1}{4} > -2x-10\end{align*}

Multiply both sides by 4: \begin{align*}4 \cdot \frac{5x-1}{4} > 4 (-2x-10)\end{align*}

Simplify: \begin{align*}5x-1 > -8x-40\end{align*}

Add \begin{align*}8x\end{align*} to both sides: \begin{align*}5x + 8x - 1 >- 8x + 8x - 40\end{align*}

Simplify: \begin{align*}13x-1>-40\end{align*}

Add 1 to both sides: \begin{align*}13x-1+1>-40+1\end{align*}

Simplify: \begin{align*}13x > -39\end{align*}

Divide both sides by 13: \begin{align*}\frac{13x}{13} > -\frac{39}{13}\end{align*}

Simplify: \begin{align*}x>-3\end{align*} Answer

Solve: \begin{align*}3x-5

### Review

Solve each multi-step inequality.

1. \begin{align*}3x-5 \ge x+2\end{align*}
2. \begin{align*}x-5 > 2x+3\end{align*}
3. \begin{align*}2(x-3) \le 3x-2\end{align*}
4. \begin{align*}3(x+1) \ge 2x+5\end{align*}
5. \begin{align*}2(x-9) \ge -1(4x+7)\end{align*}
6. \begin{align*}\frac{x}{3} < x+7\end{align*}
7. \begin{align*}\frac{x}{4} < 2x-21\end{align*}
8. \begin{align*}\frac{3(x-4)}{12} \le \frac{2x}{3}\end{align*}
9. \begin{align*}2 \left ( \frac{x}{4} + 3\right ) > 6(x-1)\end{align*}
10. \begin{align*}9x+4 \le -2 \left ( x+\frac{1}{2} \right )\end{align*}

To see the Review answers, open this PDF file and look for section 6.3.

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### Vocabulary Language: English

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