Question

2.4 puzzle time
where do young tigers swim?
write the letter of each answer in the box containing the exercise number.
solve the inequality.

1. (4x - 7 < 9)
2. (-11>10 - 7x)
3. (\frac{x}{6}+5 > 8)
4. (\frac{x}{2}+12geq14)
5. (6x - 23>25)
6. (6-\frac{x}{5}geq - 2)
7. (3geq - 3(x - 13))
8. (16 - 4x>9 - 5x)
9. (2x + 7leq2x + 8)
10. (-6(x - 1)<-14(x - 5))
11. (12x + 4x-11geq16x + 17)
12. (3(1 - x)+10xleq9(x - 2)+7)
13. the students in charge of the class booth at a carnival want to make a profit of at least $65 by selling items. they spent $55 for the materials to make the items. they will sell each item for $3. find the possible values of x, the number of items the students need to sell to reach their goal.
14. a triangle has a base of 14 centimeters and a height of ((3x - 4)) centimeters. the area of the triangle is greater than 56 centimeters. find the possible values of x.

answers
n. all real numbers
k. (xgeq7)
r. (x < 8)
e. (x>3)
o. (x < 4)
l. (x>8)
o. (xgeq40)
y. (xleq - 4)
t. (x>4)
l. (x>-7)
t. no solution
h. (xgeq12)
i. (xleq40)
t. (x>18)

Explanation:

Step1: Solve 4x - 7 < 9

Add 7 to both sides: 4x<9 + 7, 4x<16. Then divide both sides by 4, x < 4. The answer is O.

Step2: Solve - 11>10 - 7x

Add 7x to both sides: 7x-11>10. Add 11 to both sides: 7x>10 + 11, 7x>21. Divide both sides by 7, x > 3. The answer is E.

Step3: Solve $\frac{x}{6}+5>8$

Subtract 5 from both sides: $\frac{x}{6}>8 - 5$, $\frac{x}{6}>3$. Multiply both sides by 6, x>18. The answer is T.

Step4: Solve $\frac{x}{2}+12\geq14$

Subtract 12 from both sides: $\frac{x}{2}\geq14 - 12$, $\frac{x}{2}\geq2$. Multiply both sides by 2, x≥4. There is no exact match in the given answers for this one.

Step5: Solve 6x - 23>25

Add 23 to both sides: 6x>25 + 23, 6x>48. Divide both sides by 6, x>8. The answer is L.

Step6: Solve 6 - $\frac{x}{5}\geq - 2$

Subtract 6 from both sides: -$\frac{x}{5}\geq - 2-6$, -$\frac{x}{5}\geq - 8$. Multiply both sides by - 5 (and reverse the inequality sign), x≤40. The answer is I.

Step7: Solve 3≥ - 3(x - 13)

Divide both sides by - 3 (and reverse the inequality sign): - 1≤x - 13. Add 13 to both sides, x≥12. The answer is H.

Step8: Solve 16 - 4x>9 - 5x

Add 5x to both sides: 16 + x>9. Subtract 16 from both sides, x>-7. The answer is L.

Step9: Solve 2x + 7≤2x + 8

Subtract 2x from both sides: 7≤8. This is true for all real - numbers. The answer is N.

Step10: Solve - 6(x - 1)< - 14(x - 5)

Expand: - 6x+6<-14x + 70. Add 14x to both sides: 8x+6<70. Subtract 6 from both sides: 8x<64. Divide both sides by 8, x<8. The answer is R.

Step11: Solve 12x + 4x - 11≥16x + 17

Combine like terms: 16x-11≥16x + 17. Subtract 16x from both sides: - 11≥17, which is false. So there is no solution. The answer is T.

Step12: Solve 3(1 - x)+10x≤9(x - 2)+7

Expand: 3-3x + 10x≤9x-18 + 7. Combine like terms: 3 + 7x≤9x-11. Subtract 7x from both sides: 3≤2x-11. Add 11 to both sides: 14≤2x. Divide both sides by 2, x≥7. The answer is K.

Step13: Set up the inequality for the profit problem

The cost is 55, and they sell each item for 3. They want a profit of at least 65. So 3x-55≥65. Add 55 to both sides: 3x≥65 + 55, 3x≥120. Divide both sides by 3, x≥40. The answer is O.

Step14: Set up the inequality for the triangle area problem

The area of a triangle A=$\frac{1}{2}bh$. Here, b = 14 and h=(3x - 4), and A>56. So $\frac{1}{2}\times14\times(3x - 4)>56$. Simplify: 7(3x - 4)>56. Divide both sides by 7: 3x-4>8. Add 4 to both sides: 3x>12. Divide both sides by 3, x>4. The answer is T.

Answer:

1. O. x < 4
2. E. x > 3
3. T. x > 18
4. (No exact match)
5. L. x > 8
6. I. x ≤ 40
7. H. x ≥ 12
8. L. x > - 7
9. N. all real numbers
10. R. x < 8
11. T. no solution
12. K. x ≥ 7
13. O. x ≥ 40
14. T. x > 4