1) solve and graph the following inequality: 4x + 2 18
2) solve and graph the following inequality: $\frac{x}{2} \leq 6$
3) evaluate and show work: (no calculator) -156.22 + (-83.9)
4) evaluate: -12 + 44 - 16 = ?
5) solve and check for x: 4x + 12 = 60
6) simplify: |11 + (-3)| - |-4|
extended response
7) mr. smith is 9 years older than 6 times as many years old as carlos. mr. smith is 57 years old
a) write an equation that can be used to find carloss age in years. use c as your variable.
b) what is carlos age in years? show your work

Question

1) solve and graph the following inequality: 4x + 2 > 18
2) solve and graph the following inequality: $\frac{x}{2} \leq 6$
3) evaluate and show work: (no calculator) -156.22 + (-83.9)
4) evaluate: -12 + 44 - 16 = ?
5) solve and check for x: 4x + 12 = 60
6) simplify: |11 + (-3)| - |-4|
extended response
7) mr. smith is 9 years older than 6 times as many years old as carlos. mr. smith is 57 years old
a) write an equation that can be used to find carloss age in years. use c as your variable.
b) what is carlos age in years? show your work

Accepted Answer

### Problem 1: Solve and graph $ 4x + 2 > 18 $
# Explanation:
## Step 1: Subtract 2 from both sides
To isolate the term with $ x $, we subtract 2 from both sides of the inequality.
$ 4x + 2 - 2 > 18 - 2 $
Simplifying, we get $ 4x > 16 $.

## Step 2: Divide by 4
To solve for $ x $, we divide both sides of the inequality by 4.
$ \frac{4x}{4} > \frac{16}{4} $
Simplifying, we find $ x > 4 $.

## Graphing:
On a number line, we draw an open circle at 4 (since $ x $ is not equal to 4) and shade the region to the right of 4 to represent all values of $ x $ greater than 4.

# Answer:
The solution to the inequality is $ x > 4 $. The graph has an open circle at 4 and shading to the right.

### Problem 2: Solve and graph $ \frac{x}{2} \leq 6 $
# Explanation:
## Step 1: Multiply by 2
To solve for $ x $, we multiply both sides of the inequality by 2.
$ 2 \times \frac{x}{2} \leq 6 \times 2 $
Simplifying, we get $ x \leq 12 $.

## Graphing:
On a number line, we draw a closed circle at 12 (since $ x $ can be equal to 12) and shade the region to the left of 12 to represent all values of $ x $ less than or equal to 12.

# Answer:
The solution to the inequality is $ x \leq 12 $. The graph has a closed circle at 12 and shading to the left.

### Problem 3: Evaluate $ -156.22 + (-83.9) $
# Explanation:
## Step 1: Understand the addition of negative numbers
Adding a negative number is the same as subtracting its positive counterpart. So, $ -156.22 + (-83.9) = -156.22 - 83.9 $.

## Step 2: Add the absolute values
First, we find the sum of the absolute values of the two numbers: $ 156.22 + 83.9 = 240.12 $.

## Step 3: Apply the negative sign
Since both numbers are negative, the result will be negative. So, $ -156.22 + (-83.9) = -240.12 $.

# Answer:
$ -240.12 $

### Problem 4: Evaluate $ -12 + 44 - 16 $
# Explanation:
## Step 1: Add -12 and 44
First, we add $ -12 $ and $ 44 $. $ -12 + 44 = 32 $.

## Step 2: Subtract 16
Next, we subtract 16 from the result of the previous step. $ 32 - 16 = 16 $.

# Answer:
$ 16 $

### Problem 5: Solve and check $ 4x + 12 = 60 $
# Explanation:
## Step 1: Subtract 12
To isolate the term with $ x $, we subtract 12 from both sides of the equation.
$ 4x + 12 - 12 = 60 - 12 $
Simplifying, we get $ 4x = 48 $.

## Step 2: Divide by 4
To solve for $ x $, we divide both sides of the equation by 4.
$ \frac{4x}{4} = \frac{48}{4} $
Simplifying, we find $ x = 12 $.

## Checking:
Substitute $ x = 12 $ back into the original equation:
Left-hand side: $ 4(12) + 12 = 48 + 12 = 60 $
Right-hand side: $ 60 $
Since both sides are equal, $ x = 12 $ is the correct solution.

# Answer:
The solution to the equation is $ x = 12 $.

### Problem 6: Simplify $ |11 + (-3)| - |-4| $
# Explanation:
## Step 1: Simplify the expressions inside the absolute values
First, simplify $ 11 + (-3) $: $ 11 - 3 = 8 $.
Then, simplify $ |-4| $: the absolute value of -4 is 4.

## Step 2: Evaluate the absolute values
The absolute value of 8 is 8, so $ |11 + (-3)| = |8| = 8 $.
We already found that $ |-4| = 4 $.

## Step 3: Subtract
Now, subtract the two results: $ 8 - 4 = 4 $.

# Answer:
$ 4 $

### Problem 7:
#### Part A: Write an equation for Carlos's age
# Explanation:
Let $ c $ be Carlos's age in years. Mr. Smith is 9 years older than 6 times Carlos's age, and Mr. Smith is 57 years old. So, 6 times Carlos's age is $ 6c $, and adding 9 years gives Mr. Smith's age. Therefore, the equation is $ 6c + 9 = 57 $.

# Answer:
The equation is $ 6c + 9 = 57 $.

#### Part B: Find Carlos's age
# Explanation:
## Step 1: Subtract 9
To solve for $ c $, we first subtract 9 from both sides of the equation $ 6c + 9 = 57 $.
$ 6c + 9 - 9 = 57 - 9 $
Simplifying, we get $ 6c = 48 $.

## Step 2: Divide by 6
Next, we divide both sides of the equation by 6.
$ \frac{6c}{6} = \frac{48}{6} $
Simplifying, we find $ c = 8 $.

# Answer:
Carlos's age is 8 years old.

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QUESTION IMAGE

Question

Explanation:

Problem 1: Solve and graph \( 4x + 2 > 18 \)

Step 1: Subtract 2 from both sides

Step 2: Divide by 4

Graphing:

Step 1: Multiply by 2

Graphing:

Step 1: Understand the addition of negative numbers

Step 2: Add the absolute values

Step 3: Apply the negative sign

Answer:

Problem 2: Solve and graph \( \frac{x}{2} \leq 6 \)