Question

show all work as you solve each inequality below.

1. $4.5k + 18 - 24.1 - 1.3k \geq 14.7$
2. $\frac{3}{4}(-x + 12) < 18$
3. $24 - 7.2(-4x + 3) \leq -26.4$
4. rodney is flying on an airplane to puerto rico. his suitcase and the contents inside must weigh less than 50lbs. his suitcase weighs 4 pounds and the contents he wants to pack weigh 49 pounds. if each of his shirts weighs 0.75 pounds, write an inequality to represent the number of shirts he needs to remove to meet the flight requirements.
5. selene and bennett are solving the inequality below. selene says you will need to flip the inequality sign while bennett claims you will not need to flip it. who is correct? explain your reasoning.

$17 - 5(2x + 1) + 12x \leq -22$
roll a number cube and solve the corresponding inequality for extra practice.

roll	inequality	solution
2	$\frac{1}{2}m + 2(m - 3) \leq 13$
3	$5(-4x + 1) < 45$
4	$s + 4 - 7s - 9 \geq 31$
5	$16 > 11 - 2(x - 4)$
6	$\frac{1}{3}(x - 27) + \frac{1}{3}x > 31$

summarize today’s lesson:

Explanation:

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Problem 4

Step1: Combine like terms

$4.5k - 1.3k + 18 - 24.1 \geq 14.7$
$3.2k - 6.1 \geq 14.7$

Step2: Add 6.1 to both sides

$3.2k - 6.1 + 6.1 \geq 14.7 + 6.1$
$3.2k \geq 20.8$

Step3: Divide by 3.2

$k \geq \frac{20.8}{3.2}$
$k \geq 6.5$

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Problem 5

Step1: Multiply both sides by $\frac{4}{3}$

$\frac{4}{3} \times \frac{3}{4}(-x + 12) < 18 \times \frac{4}{3}$
$-x + 12 < 24$

Step2: Subtract 12 from both sides

$-x + 12 - 12 < 24 - 12$
$-x < 12$

Step3: Multiply by -1 (flip sign)

$x > -12$

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Problem 6

Step1: Distribute -7.2

$24 + 28.8x - 21.6 \leq -26.4$

Step2: Combine constants

$28.8x + 2.4 \leq -26.4$

Step3: Subtract 2.4

$28.8x \leq -26.4 - 2.4$
$28.8x \leq -28.8$

Step4: Divide by 28.8

$x \leq -1$

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Problem 7

Step1: Define variable, set up inequality

Let $s$ = number of shirts to remove. Total weight: $4 + (49 - 0.75s) < 50$

Step2: Simplify left side

$53 - 0.75s < 50$

Step3: Isolate term with $s$

$-0.75s < 50 - 53$
$-0.75s < -3$

Step4: Divide by -0.75 (flip sign)

$s > \frac{-3}{-0.75}$
$s > 4$

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Problem 8

Step1: Distribute -5

$17 - 10x - 5 + 12x \leq -22$

Step2: Combine like terms

$2x + 12 \leq -22$

Step3: Subtract 12

$2x \leq -22 - 12$
$2x \leq -34$

Step4: Divide by 2

$x \leq -17$
No multiplication/division by negative number, so no sign flip.

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Extra Practice (Roll 1-6)

Roll 1

Step1: Simplify left side

$-x - 9 + 24 > 17$
$-x + 15 > 17$

Step2: Isolate $-x$

$-x > 17 - 15$
$-x > 2$

Step3: Multiply by -1 (flip sign)

$x < -2$

Roll 2

Step1: Distribute 2, combine terms

$\frac{1}{2}m + 2m - 6 \leq 13$
$\frac{5}{2}m - 6 \leq 13$

Step2: Add 6

$\frac{5}{2}m \leq 19$

Step3: Multiply by $\frac{2}{5}$

$m \leq 19 \times \frac{2}{5}$
$m \leq \frac{38}{5}$
$m \leq 7.6$

Roll 3

Step1: Distribute 5

$-20x + 5 < 45$

Step2: Subtract 5

$-20x < 45 - 5$
$-20x < 40$

Step3: Divide by -20 (flip sign)

$x > -2$

Roll 4

Step1: Combine like terms

$-6s - 5 \geq 31$

Step2: Add 5

$-6s \geq 31 + 5$
$-6s \geq 36$

Step3: Divide by -6 (flip sign)

$s \leq -6$

Roll 5

Step1: Distribute -2

$16 > 11 - 2x + 8$

Step2: Combine constants

$16 > 19 - 2x$

Step3: Subtract 19

$16 - 19 > -2x$
$-3 > -2x$

Step4: Divide by -2 (flip sign)

$\frac{-3}{-2} < x$
$x > 1.5$

Roll 6

Step1: Distribute $\frac{1}{3}$

$\frac{1}{3}x - 9 + \frac{1}{3}x > 31$

Step2: Combine like terms

$\frac{2}{3}x - 9 > 31$

Step3: Add 9

$\frac{2}{3}x > 31 + 9$
$\frac{2}{3}x > 40$

Step4: Multiply by $\frac{3}{2}$

$x > 40 \times \frac{3}{2}$
$x > 60$

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Answer:

1. Problem 4: $\boldsymbol{k \geq 6.5}$
2. Problem 5: $\boldsymbol{x > -12}$
3. Problem 6: $\boldsymbol{x \leq -1}$
4. Problem 7: $\boldsymbol{s > 4}$ (Rodney must remove more than 4 shirts)
5. Problem 8: Bennett is correct. The final step only involves dividing by a positive number, so the inequality sign does not need to be flipped. Solution: $\boldsymbol{x \leq -17}$
6. Extra Practice:

• Roll 1: $\boldsymbol{x < -2}$
• Roll 2: $\boldsymbol{m \leq 7.6}$
• Roll 3: $\boldsymbol{x > -2}$
• Roll 4: $\boldsymbol{s \leq -6}$
• Roll 5: $\boldsymbol{x > 1.5}$
• Roll 6: $\boldsymbol{x > 60}$