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|
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 carlos’s age in years. use c as your variable.

Explanation:

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

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

Now, divide both sides by 4 to solve for \( x \).
\( \frac{4x}{4} > \frac{16}{4} \)
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 \( 4 \).

Step 1: Multiply by 2

To solve for \( x \), 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 \( 12 \).

Step 1: Add the absolute values

When adding two negative numbers, we add their absolute values and keep the negative sign.
The absolute value of \( -156.22 \) is \( 156.22 \), and the absolute value of \( -83.9 \) is \( 83.9 \).
Adding these: \( 156.22 + 83.9 = 240.12 \).

Step 2: Apply the negative sign

Since both numbers are negative, the result is negative.
Thus, \( -156.22 + (-83.9) = -240.12 \).

Answer:

\( x > 4 \) (Graph: Open circle at 4, shade right)

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