Question

7-84. evaluate the inequality ( 3x - 3 geq 2x + 3 ) for the following listed values of ( x ). decide if the value makes the statement true or false. show your work.
a. ( x = -3 )
b. ( x = 9.5 )
c. ( x = 6 )
d. ( x = 10\frac{1}{2} )
7-85. mr. takaya can eat three slices of pizza in five minutes. if he continues to eat at the same rate, how long will it take him to eat the whole pizza, which has twelve slices? how many slices could he eat in half of an hour?

Explanation:

First Problem (7-83: Rectangular Lawn Dimensions & Area)

Step1: Define variables for width/length

Let width = $w$. Length $l = 3w + 3$.

Step2: Set up perimeter equation

Perimeter formula: $P=2(l+w)$. Substitute $P=294$, $l=3w+3$:
$294 = 2((3w+3)+w)$

Step3: Simplify and solve for $w$

Simplify inside parentheses: $294 = 2(4w+3)$
Divide both sides by 2: $147 = 4w+3$
Subtract 3: $144 = 4w$
Solve for $w$: $w = \frac{144}{4} = 36$

Step4: Calculate length $l$

Substitute $w=36$: $l = 3(36)+3 = 111$

Step5: Calculate area of the lawn

Area formula: $A = l \times w$
$A = 111 \times 36$

Inequality: $3x - 3 \geq 2x + 3$, simplify to $x \geq 6$ (by subtracting $2x$ and adding 3 to both sides). Test each value:

Step1: Test $x=-3$

Substitute into inequality: $3(-3)-3 \geq 2(-3)+3$
Calculate: $-9-3 \geq -6+3 \implies -12 \geq -3$

Step2: Test $x=9.5$

Substitute into inequality: $3(9.5)-3 \geq 2(9.5)+3$
Calculate: $28.5-3 \geq 19+3 \implies 25.5 \geq 22$

Step3: Test $x=6$

Substitute into inequality: $3(6)-3 \geq 2(6)+3$
Calculate: $18-3 \geq 12+3 \implies 15 \geq 15$

Step4: Test $x=10\frac{1}{3}$

Substitute $x=\frac{31}{3}$: $3(\frac{31}{3})-3 \geq 2(\frac{31}{3})+3$
Calculate: $31-3 \geq \frac{62}{3}+3 \implies 28 \geq \frac{71}{3} \approx 23.67$

Step1: Find rate of eating

Rate = $\frac{3 \text{ slices}}{5 \text{ minutes}} = 0.6$ slices per minute

Step2: Time to eat 12 slices

Time = $\frac{\text{Total slices}}{\text{Rate}} = \frac{12}{0.6}$

Step3: Slices eaten in 30 minutes

Slices = $\text{Rate} \times \text{Time} = 0.6 \times 30$

Answer:

a. Width = 36 feet, Length = 111 feet
b. Area = 3996 square feet

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Second Problem (7-84: Inequality Evaluation)