31. solve: |x - 5| = 9
32. solve: -4(x + 3) + 2x = -18
33. write the equation of a vertical line through (-2, 7)
34. write the equation of a horizontal line through (0, -6)
35. find x- and y-intercepts of 3x - 4y = 24
36. graph 2x + 3y = 12 using intercepts method (label both intercepts)
37. write the equation in slope-intercept form: passes through (0, -5), slope = 3

Question

Accepted Answer

### Problem 31: Solve $|x - 5| = 9$
# Explanation:
## Step 1: Recall absolute value definition
The absolute value equation $|A| = B$ (where $B \geq 0$) implies $A = B$ or $A = -B$. So for $|x - 5| = 9$, we have two cases:
Case 1: $x - 5 = 9$
Case 2: $x - 5 = -9$

## Step 2: Solve Case 1
For $x - 5 = 9$, add 5 to both sides:
$x - 5 + 5 = 9 + 5$
$x = 14$

## Step 3: Solve Case 2
For $x - 5 = -9$, add 5 to both sides:
$x - 5 + 5 = -9 + 5$
$x = -4$

# Answer: $x = 14$ or $x = -4$

### Problem 32: Solve $-4(x + 3) + 2x = -18$
# Explanation:
## Step 1: Distribute the -4
$-4(x + 3) + 2x = -18$ becomes $-4x - 12 + 2x = -18$

## Step 2: Combine like terms
Combine $-4x$ and $2x$: $-2x - 12 = -18$

## Step 3: Add 12 to both sides
$-2x - 12 + 12 = -18 + 12$
$-2x = -6$

## Step 4: Divide by -2
$\frac{-2x}{-2} = \frac{-6}{-2}$
$x = 3$

# Answer: $x = 3$

### Problem 33: Write the equation of a vertical line through $(-2, 7)$
# Explanation:
## Step 1: Recall vertical line equation
A vertical line has an undefined slope and its equation is of the form $x = a$, where $a$ is the x - coordinate of any point on the line.

## Step 2: Determine the value of $a$
The point $(-2, 7)$ has an x - coordinate of $-2$. So the equation of the vertical line is $x = -2$.

# Answer: $x = -2$

### Problem 34: Write the equation of a horizontal line through $(0, -6)$
# Explanation:
## Step 1: Recall horizontal line equation
A horizontal line has a slope of 0 and its equation is of the form $y = b$, where $b$ is the y - coordinate of any point on the line.

## Step 2: Determine the value of $b$
The point $(0, -6)$ has a y - coordinate of $-6$. So the equation of the horizontal line is $y = -6$.

# Answer: $y = -6$

### Problem 35: Find x - and y - intercepts of $3x - 4y = 24$
# Explanation:
## Step 1: Find x - intercept
The x - intercept is the point where $y = 0$. Substitute $y = 0$ into the equation $3x - 4y = 24$:
$3x - 4(0) = 24$
$3x = 24$
Divide both sides by 3: $\frac{3x}{3}=\frac{24}{3}$
$x = 8$
So the x - intercept is $(8, 0)$

## Step 2: Find y - intercept
The y - intercept is the point where $x = 0$. Substitute $x = 0$ into the equation $3x - 4y = 24$:
$3(0)-4y = 24$
$-4y = 24$
Divide both sides by - 4: $\frac{-4y}{-4}=\frac{24}{-4}$
$y=-6$
So the y - intercept is $(0, -6)$

# Answer: x - intercept: $(8, 0)$, y - intercept: $(0, -6)$

### Problem 36: Graph $2x + 3y = 12$ using intercepts method
# Explanation:
## Step 1: Find x - intercept
Set $y = 0$ in $2x + 3y = 12$:
$2x+3(0)=12$
$2x = 12$
Divide by 2: $x = 6$
So the x - intercept is $(6, 0)$

## Step 2: Find y - intercept
Set $x = 0$ in $2x + 3y = 12$:
$2(0)+3y = 12$
$3y = 12$
Divide by 3: $y = 4$
So the y - intercept is $(0, 4)$

## Step 3: Graph the line
Plot the points $(6, 0)$ (x - intercept) and $(0, 4)$ (y - intercept) on the coordinate plane and draw a straight line through them. Label the x - intercept as $(6, 0)$ and the y - intercept as $(0, 4)$.

# Answer: x - intercept $(6, 0)$, y - intercept $(0, 4)$; line passes through these two points.

### Problem 37: Write the equation in slope - intercept form: passes through $(0, -5)$, slope $= 3$
# Explanation:
## Step 1: Recall slope - intercept form
The slope - intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept.

## Step 2: Identify $m$ and $b$
We know the slope $m = 3$. The line passes through $(0, -5)$, which is the y - intercept (since when $x = 0$, $y = b$). So $b=-5$.

## Step 3: Substitute $m$ and $b$ into the formula
Substitute $m = 3$ and $b=-5$ into $y = mx + b$:
$y = 3x-5$

# Answer: $y = 3x - 5$

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QUESTION IMAGE

Question

Explanation:

Problem 31: Solve \(|x - 5| = 9\)

Step 1: Recall absolute value definition

Step 2: Solve Case 1

Step 3: Solve Case 2

Step 1: Distribute the -4

Step 2: Combine like terms

Step 3: Add 12 to both sides

Step 4: Divide by -2

Step 1: Recall vertical line equation

Step 2: Determine the value of \(a\)

Answer:

Problem 32: Solve \(-4(x + 3) + 2x = -18\)