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Multi-Step Inequalities

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Solving Multi-Step Inequalities
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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 added enough antifreeze to lower the freezing point of the liquids to at most -35^\circ C (in Celsius). Write the inequality in Fahrenheit.

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Khan Academy: Multi-Step Inequalities 2

Guidance

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.

Example A

Is x = -3 a solution to 2(3x-5) \le x+10 ?

Solution: Plug in -3 for x and see if the inequality is true.

2(3(-3)-5) & \le (-3)+10\\2(-9-5) & \le 7\\2 \cdot -14 & \le 7\\-28 & \le 7

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

Example B

Solve and graph the inequality from Example A.

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

2(3x-5) & \le x+10\\6x-10 & \le x+10

Now, subtract the x on the right side to move it to the left side of the inequality. You can also add the 10’s together and solve.

& \ 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

Test a solution, x = 0: 2(3(0)-5) \le 0+10 \checkmark

-10 \le 10

The graph looks like:

Example C

Solve 8x-5-4x \ge 37-2x

Solution: First, combine like terms on the left side. Then, solve for x .

& 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

Test a solution, x = 10: 8(10)-5-4(10) \ge 37-2(10)

80-5-40 \ge 37-20

35 \ge 17 \checkmark

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

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

So, the temperature can be equal to or higher than -31^\circ F .

Guided Practice

1. Is x = 12 a solution to -3(x-10)+18 \ge x-25 ?

Solve and graph the following inequalities.

2. -(x+16)+3x>8

3. 24-9x<6x-21

Answers

1. Plug in 12 for x and simplify.

-3(12-10)+18 &\ge 12-35\\-3 \cdot 2 + 18 & \ge -13\\-6+18 & \ge -13

This is true because 12 \ge -13 , so 12 is a solution.

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

& -(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

Test a solution, x = 15:

-(15+16)+3(15) &> 8\\-31+45 &> 8 \\14 &> 8

3. First, add 9x to both sides and add 21 to both sides.

& \ 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<x

Test a solution, x = 10:

24-9(10) &< 6(10)-21\\24-90 &< 60-21 \\-66 &< 39

Practice

Determine if the following numbers are solutions to -7(2x-5)+12>-4x-13 .

  1. x=4
  2. x=10
  3. x=6

Solve and graph the following inequalities.

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

Solve the following inequalities.

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

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