Question

solve the inequality and graph the solution.
12 ≥ 3(k - 4)
to draw a ray, plot an endpoint and select an arrow. select an endpoint to change it from closed to open. select the middle of the ray to delete it.
-10 -8 -6 -4 -2 0 2 4 6 8 10

Explanation:

Step1: Divide both sides by 3

To isolate the term with \( k \), we divide both sides of the inequality \( 12 \geq 3(k - 4) \) by 3. Since 3 is positive, the direction of the inequality sign remains the same.

$$ \frac{12}{3} \geq \frac{3(k - 4)}{3} $$

Simplifying both sides, we get:

$$ 4 \geq k - 4 $$

Step2: Add 4 to both sides

To solve for \( k \), we add 4 to both sides of the inequality \( 4 \geq k - 4 \).

$$ 4 + 4 \geq k - 4 + 4 $$

Simplifying both sides, we find:

$$ 8 \geq k $$

Which can also be written as \( k \leq 8 \).

Graphing the Solution:

• The solution \( k \leq 8 \) means we include all real numbers less than or equal to 8.
• On the number line, we plot a closed dot at 8 (because the inequality is "less than or equal to," so 8 is included in the solution).
• Then we draw an arrow pointing to the left (towards negative infinity) to represent all numbers less than 8.

Answer:

The solution to the inequality \( 12 \geq 3(k - 4) \) is \( k \leq 8 \). To graph it, place a closed dot at 8 on the number line and draw an arrow to the left.