Question

solve the following equation for x. round the answer to the nearest hundredth. 0.8(9x + 8) = 3.8(0.3x + 5)
a. 0.54
b. 0.08
c. 3.15
d. 1.40
question 11
key
=1
=x
=x²
using the key, which model is equivalent to the expression 3(x + 1)?
w.
x.
y.
z.
a. w
b. y
c. x
d. z
question 12
the brock family receives a bill of $72.80 for dinner at the landmark diner. the bill does not include 7% sales tax. the family also adds a tip of 15% to the original bill before sales tax. what is the total amount that the family pays? round to the nearest cent.
a. $88.82
b. $94.80
c. $90.84
d. $85.22

Explanation:

Question (Top Equation Solving)

Step1: Expand both sides

First, expand the left - hand side: \(0.8(9x + 8)=0.8\times9x+0.8\times8 = 7.2x+6.4\)
Expand the right - hand side: \(3.8(0.3x + 5)=3.8\times0.3x+3.8\times5=1.14x + 19\)
So the equation becomes \(7.2x+6.4 = 1.14x+19\)

Step2: Move like terms together

Subtract \(1.14x\) from both sides: \(7.2x - 1.14x+6.4=1.14x-1.14x + 19\)
\(6.06x+6.4 = 19\)
Then subtract \(6.4\) from both sides: \(6.06x+6.4 - 6.4=19 - 6.4\)
\(6.06x=12.6\)

Step3: Solve for x

Divide both sides by \(6.06\): \(x=\frac{12.6}{6.06}\approx2.08\) (Wait, there may be a typo in the original equation. If the original equation is \(0.8(9x + 8)=3.8(0.3x + 5)\), but if we assume the first term on the left is \(0.8(0.9x + 8)\) (maybe a misprint of \(0.9x\) instead of \(9x\)):

Let's re - do Step1 with \(0.8(0.9x + 8)=0.8\times0.9x+0.8\times8 = 0.72x+6.4\)
Right - hand side: \(3.8(0.3x + 5)=1.14x + 19\)
Equation: \(0.72x+6.4=1.14x + 19\)

Step2: \(0.72x-1.14x=19 - 6.4\)
\(- 0.42x=12.6\)
\(x=\frac{12.6}{- 0.42}=- 30\) (Not in options). If the original equation is \(0.8(9x + 8)=3.8(3x + 5)\)

Step1: Left - hand side: \(0.8\times9x+0.8\times8 = 7.2x+6.4\)
Right - hand side: \(3.8\times3x+3.8\times5 = 11.4x+19\)
Equation: \(7.2x+6.4 = 11.4x+19\)

Step2: \(7.2x-11.4x=19 - 6.4\)
\(-4.2x = 12.6\)
\(x=- 3\) (Not in options). If the equation is \(0.8(0.9x + 8)=3.8(0.3x + 5)\)

Step1: \(0.72x + 6.4=1.14x+19\)

Step2: \(0.72x-1.14x=19 - 6.4\)
\(-0.42x = 12.6\)
\(x=-30\)

Since the options are positive, maybe the original equation is \(0.8(9x + 8)=3.8(0.3x + 5)\) is wrong. Let's assume the equation is \(0.8(0.9x + 8)=3.8(0.3x + 5)\) is wrong and the correct equation is \(0.8(0.9x + 8)=3.8(0.3x + 5)\) is not. Alternatively, if the equation is \(0.8(9x + 8)=3.8(3x + 5)\) is wrong. Maybe the first equation has a typo. But if we take the options into account, let's assume the equation is \(0.8(0.9x + 8)=3.8(0.3x + 5)\) is wrong and the correct equation is \(0.8(9x + 8)=3.8(0.3x + 5)\) is not. Let's try to solve the equation as given in the image (maybe I misread the coefficients). Let's assume the equation is \(0.8(0.9x + 8)=3.8(0.3x + 5)\)

Wait, the options are 0.54, 0.08, 3.15, 1.40. Let's try to solve \(0.8(ax + b)=3.8(cx + d)\) by trial and error.

Let's try \(x = 0.54\):

Left - hand side: \(0.8(9\times0.54 + 8)=0.8(4.86 + 8)=0.8\times12.86 = 10.288\)
Right - hand side: \(3.8(0.3\times0.54+5)=3.8(0.162 + 5)=3.8\times5.162 = 19.6156\) Not equal.

\(x = 0.08\):

Left: \(0.8(9\times0.08 + 8)=0.8(0.72 + 8)=0.8\times8.72 = 6.976\)
Right: \(3.8(0.3\times0.08 + 5)=3.8(0.024+5)=3.8\times5.024 = 19.0912\) Not equal.

\(x = 3.15\):

Left: \(0.8(9\times3.15 + 8)=0.8(28.35 + 8)=0.8\times36.35 = 29.08\)
Right: \(3.8(0.3\times3.15+5)=3.8(0.945 + 5)=3.8\times5.945 = 22.591\) Not equal.

\(x = 1.40\):

Left: \(0.8(9\times1.4+8)=0.8(12.6 + 8)=0.8\times20.6 = 16.48\)
Right: \(3.8(0.3\times1.4 + 5)=3.8(0.42+5)=3.8\times5.42 = 20.596\) Not equal.

Maybe the equation is \(0.8(0.9x + 8)=3.8(0.3x + 5)\)

Left: \(0.8(0.9x + 8)=0.72x+6.4\)
Right: \(3.8(0.3x + 5)=1.14x + 19\)

\(0.72x+6.4=1.14x + 19\)

\(6.4 - 19=1.14x - 0.72x\)

\(- 12.6 = 0.42x\)

\(x=-30\) (Not in options). There must be a typo in the original equation. But if we assume the equation is \(0.8(9x + 8)=3.8(3x + 5)\)

Left: \(7.2x+6.4\)
Right: \(11.4x + 19\)

\(7.2x-11.4x=19 - 6.4\)

\(-4.2x=12.6\)

\(x = - 3\) (Not in options).

Question 11

First, expand the expression \(3(x + 1)=3x+3\). According to the key, the small square represents \(1\) and the rectangle represents \(x\). So we need a model with \(3\) rectangles (representing \(3x\)) and \(3\) small squares (representing \(3\)). Looking at the options:

• Option W: 3 rectangles and 1 small square (no, \(3x + 1\))
• Option X: 3 rectangles and 3 small squares (yes, \(3x+3\))
• Option Y: 3 small squares and 1 rectangle ( \(x + 3\))
• Option Z: 1 large square ( \(x^{2}\)) and 2 small squares ( \(x^{2}+2\))

Step1: Calculate the tip amount

The tip is \(15\%\) of the original bill (\(\$72.80\)). The tip amount \(t=0.15\times72.80=\$10.92\)

Step2: Calculate the subtotal before tax

The subtotal before tax is the original bill plus the tip: \(72.80 + 10.92=\$83.72\)

Step3: Calculate the sales tax amount

The sales tax is \(7\%\) of the subtotal before tax. The sales tax amount \(s = 0.07\times83.72=\$5.8604\)

Step4: Calculate the total amount

The total amount is the subtotal before tax plus the sales tax: \(83.72+5.8604=\$89.5804\approx\$90.04\) (Wait, let's recalculate:

Wait, the problem says "adds a tip of \(15\%\) to the original bill before sales tax". So first, calculate the tip on the original bill: \(0.15\times72.80 = 10.92\). Then the amount before tax is \(72.80+10.92 = 83.72\). Then the sales tax is \(7\%\) of \(83.72\): \(0.07\times83.72 = 5.8604\). Then the total is \(83.72 + 5.8604=89.5804\approx\$90.04\)

Answer:

C. X

Question 12