Question

1. determine whether (y = - 2x^{2}) is a function. if so, state the domain and range.

simplify.

1. (\frac{15x^{3}y^{2}}{5xy})
2. (\frac{(m^{3}n^{2})^{4}(mn^{3})^{2}}{(m^{2}n^{4})^{3}mn})

solve each equation.

1. (0.02x+0.01 = 0.17)
2. (4\frac{1}{3}x-\frac{1}{2}=1\frac{2}{3})

solve and graph each inequality.

1. (2x + 3geq4x - 5)
2. (|x - 2|>6)
3. real - world: the amount of water added to a dry mortar mix varies directly with the amount of dry mortar. 100 lb of dry mortar requires 11.25 pt of water.

a. find the constant of variation for the dry mortar mix.
b. find how many pints of water are needed for 240 lb of dry mortar mix.

1. simplify (43(x + 9)+2).
2. graphing calculator: find each product. check using the graphing calculator.
3. (\begin{bmatrix}1&0\\0&1end{bmatrix}\times\begin{bmatrix}1&-3\\-2&2end{bmatrix})
4. (\begin{bmatrix}0&-1\\-1&0end{bmatrix}\times\begin{bmatrix}4&-1\\-2&-6end{bmatrix}
5. find each percent change. tell whether it is a percent increase or decrease.
6. 265 to 318
7. 17 to 14.45
8. estimate: one cubic foot of pennies contains approximately (4.9\times10^{4}) pennies. think about covering the entire earth with two layers of pennies. the total number of pennies needed could be stacked in a cube that measures (2.73\times10^{4}) ft on each side. approximately how many pennies are in this cube?
9. error analysis: look at the explanation for graphing the solution of two inequalities. to draw the graph of the solution for (xleq2) or (x > 5) you draw a number line. draw an open circle around the point 2 and a line through all the values less than 2. next draw an open circle around the point 5 and a line through all values greater than 5. is this explanation correct?

Explanation:

Step1: Analyze problem 6

Solve the equation \(4\frac{1}{3}x-\frac{1}{2}=1\frac{2}{3}\). First, convert mixed - numbers to improper fractions.
\(4\frac{1}{3}=\frac{13}{3}\) and \(1\frac{2}{3}=\frac{5}{3}\). The equation becomes \(\frac{13}{3}x-\frac{1}{2}=\frac{5}{3}\).

Step2: Add \(\frac{1}{2}\) to both sides

\(\frac{13}{3}x=\frac{5}{3}+\frac{1}{2}\). Find a common denominator, which is 6. So \(\frac{5}{3}+\frac{1}{2}=\frac{10 + 3}{6}=\frac{13}{6}\). Then \(\frac{13}{3}x=\frac{13}{6}\).

Step3: Solve for \(x\)

Multiply both sides by the reciprocal of \(\frac{13}{3}\), which is \(\frac{3}{13}\). \(x=\frac{13}{6}\times\frac{3}{13}=\frac{1}{2}\).

Step4: Analyze problem 8

Solve the inequality \(2x + 3\geq4x-5\).

Step5: Subtract \(2x\) from both sides

\(3\geq2x - 5\).

Step6: Add 5 to both sides

\(8\geq2x\).

Step7: Divide both sides by 2

\(4\geq x\) or \(x\leq4\).

Step8: Analyze problem 9a

If the amount of water \(w\) added to a dry - mortar mix varies directly with the amount of dry mortar \(m\), the equation is \(w = km\), where \(k\) is the constant of variation. Given that \(m = 100\) lb and \(w=11.25\) pt, then \(k=\frac{w}{m}=\frac{11.25}{100}=0.1125\).

Step9: Analyze problem 9b

Using \(w = km\) with \(k = 0.1125\) and \(m = 240\) lb, then \(w=0.1125\times240 = 27\) pt.

Answer:

1. \(x=\frac{1}{2}\)
2. \(x\leq4\)

9a. The constant of variation \(k = 0.1125\)
9b. 27 pt of water are needed.