solve and graph each inequality.
*7. 3(x - 4)6
*8. 2x + 3≥4x - 5
9. multi - step 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.
10. simplify 43(x + 9)+2.
graphing calculator find each product. check using the graphing calculator.
*11. $\begin{bmatrix}1&0\\0&1\end{bmatrix}\times\begin{bmatrix}-1&2\\3&-2\end{bmatrix}$
*12. $\begin{bmatrix}0&-1\\-1&0\end{bmatrix}\times\begin{bmatrix}4&-1\\-2&-6\end{bmatrix}$
find each percent change. tell whether it is a percent increase or decrease.
13. 265 to 318
14. 17 to 14.45
15. estimate one cubic foot of pennies contains approximately 4.9×10^4 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 2.73×10^4 ft on each side. approximately how many pennies are in this cube?
*16. error analysis look at the explanation given for graphing the solution of two inequalities. to draw the graph of the solution x≤2 or x5 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?

Question

solve and graph each inequality.
*7. 3(x - 4)>6
*8. 2x + 3≥4x - 5
9. multi - step 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.
10. simplify 43(x + 9)+2.
graphing calculator find each product. check using the graphing calculator.
*11. $\begin{bmatrix}1&0\\0&1\end{bmatrix}\times\begin{bmatrix}-1&2\\3&-2\end{bmatrix}$
*12. $\begin{bmatrix}0&-1\\-1&0\end{bmatrix}\times\begin{bmatrix}4&-1\\-2&-6\end{bmatrix}$
find each percent change. tell whether it is a percent increase or decrease.
13. 265 to 318
14. 17 to 14.45
15. estimate one cubic foot of pennies contains approximately 4.9×10^4 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 2.73×10^4 ft on each side. approximately how many pennies are in this cube?
*16. error analysis look at the explanation given for graphing the solution of two inequalities. to draw the graph of the solution x≤2 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?

Accepted Answer

### 7. Solve $3(x - 4)>6$
# Explanation:
## Step1: Distribute the 3
$$3x-12 > 6$$
## Step2: Add 12 to both sides
$$3x-12 + 12>6 + 12$$
$$3x>18$$
## Step3: Divide both sides by 3
$$\frac{3x}{3}>\frac{18}{3}$$
$$x > 6$$

### 8. Solve $2x+3\geq4x - 5$
# Explanation:
## Step1: Subtract $2x$ from both sides
$$2x+3-2x\geq4x - 5-2x$$
$$3\geq2x - 5$$
## Step2: Add 5 to both sides
$$3 + 5\geq2x-5 + 5$$
$$8\geq2x$$
## Step3: Divide both sides by 2
$$\frac{8}{2}\geq\frac{2x}{2}$$
$$4\geq x$$ or $x\leq4$

### 9.
a. If the amount of water $w$ added to a dry mortar mix varies directly with the amount of dry mortar $m$, and $w = 11.25$ pt when $m = 100$ lb, the equation for direct - variation is $w=km$. First, find the constant of variation $k$.
# Explanation:
## Step1: Substitute $w = 11.25$ and $m = 100$ into $w=km$
$$11.25=k\times100$$
## Step2: Solve for $k$
$$k=\frac{11.25}{100}=0.1125$$
The constant of variation is $0.1125$ pt/lb.

b. Now, find $w$ when $m = 240$ lb.
# Explanation:
## Step1: Use the equation $w = km$ with $k = 0.1125$ and $m = 240$
$$w=0.1125\times240$$
$$w = 27$$ pt

### 10. Simplify $4[3(x + 9)+2]$
# Explanation:
## Step1: First, simplify the expression inside the brackets
$$3(x + 9)+2=3x+27 + 2=3x+29$$
## Step2: Then, multiply by 4
$$4(3x + 29)=12x+116$$

### 11. Calculate $\begin{bmatrix}1&0\0&1\end{bmatrix}\times\begin{bmatrix}-1&2\3&-2\end{bmatrix}$
# Explanation:
## Step1: Use the rule for matrix - multiplication. The element in the $i$ - th row and $j$ - th column of the product matrix is the dot - product of the $i$ - th row of the first matrix and the $j$ - th column of the second matrix.
The first element of the product matrix: $(1\times(-1))+(0\times3)=-1$
The second element of the first row: $(1\times2)+(0\times(-2)) = 2$
The first element of the second row: $(0\times(-1))+(1\times3)=3$
The second element of the second row: $(0\times2)+(1\times(-2))=-2$
The product is $\begin{bmatrix}-1&2\3&-2\end{bmatrix}$

### 12. Calculate $\begin{bmatrix}0&-1\-1&0\end{bmatrix}\times\begin{bmatrix}4&-1\-2&-6\end{bmatrix}$
# Explanation:
## Step1: Use the rule for matrix - multiplication.
The first element of the product matrix: $(0\times4)+(-1\times(-2)) = 2$
The second element of the first row: $(0\times(-1))+(-1\times(-6))=6$
The first element of the second row: $(-1\times4)+(0\times(-2))=-4$
The second element of the second row: $(-1\times(-1))+(0\times(-6)) = 1$
The product is $\begin{bmatrix}2&6\-4&1\end{bmatrix}$

### 13. Find the percent change from 265 to 318
# Explanation:
## Step1: Use the percent - change formula $\text{Percent Change}=\frac{\text{New Value}-\text{Old Value}}{\text{Old Value}}\times100\%$
$\text{Percent Change}=\frac{318 - 265}{265}\times100\%=\frac{53}{265}\times100\%$
## Step2: Simplify the fraction and multiply by 100%
$\frac{53}{265}=0.2$, so $0.2\times100\% = 20\%$
Since the new value is greater than the old value, it is a percent increase.

### 14. Find the percent change from 17 to 14.45
# Explanation:
## Step1: Use the percent - change formula $\text{Percent Change}=\frac{\text{New Value}-\text{Old Value}}{\text{Old Value}}\times100\%$
$\text{Percent Change}=\frac{14.45 - 17}{17}\times100\%=\frac{- 2.55}{17}\times100\%$
## Step2: Simplify the fraction and multiply by 100%
$\frac{-2.55}{17}=-0.15$, so $-0.15\times100\%=-15\%$
Since the new value is less than the old value, it is a percent decrease.

### 15. This is a very complex estimation problem. But a rough estimate:
The surface area of a cube with side length $s = 2.73\times10^{4}$ ft is $6s^{2}=6\times(2.73\times10^{4})^{2}=6\times7.4529\times10^{8}\text{ ft}^2$. One cubic foot has $4.9\times10^{4}$ pennies. Two - layer coverage would require a very large number of pennies, but without more information about the packing efficiency etc., a full - fledged answer is difficult. However, if we just consider the volume of the cube for a very rough estimate, the volume of the cube $V=s^{3}=(2.73\times10^{4})^{3}=2.03\times10^{13}\text{ ft}^3$. The number of pennies for two - layer coverage (very rough) would be $2\times4.9\times10^{4}\times6\times(2.73\times10^{4})^{2}$ (assuming we can spread them out evenly on the cube's surface).

### 16. The explanation is incorrect.
For $x\leq2$ or $x > 5$, when graphing $x\leq2$, we draw a closed circle at $x = 2$ (because the inequality includes equality) and a line through all values less than 2. When graphing $x>5$, we draw an open circle at $x = 5$ and a line through all values greater than 5.

# Answer:
7. $x>6$
8. $x\leq4$
9. a. $k = 0.1125$ pt/lb; b. 27 pt
10. $12x + 116$
11. $\begin{bmatrix}-1&2\3&-2\end{bmatrix}$
12. $\begin{bmatrix}2&6\-4&1\end{bmatrix}$
13. 20%, percent increase
14. 15%, percent decrease
15. Rough estimate depends on more factors about packing, but volume - based rough estimate can be made as shown above
16. Incorrect. Closed circle at 2 for $x\leq2$ and open circle at 5 for $x > 5$ when graphing.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

7. Solve \(3(x - 4)>6\)

Step1: Distribute the 3

Step2: Add 12 to both sides

Step3: Divide both sides by 3

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

Step1: Subtract \(2x\) from both sides

Step2: Add 5 to both sides

Step3: Divide both sides by 2

9.

Step1: Substitute \(w = 11.25\) and \(m = 100\) into \(w=km\)

Step2: Solve for \(k\)

Answer: