Question

1. which graph is the solution to the inequality 6.4 - 4x ≥ -2.8? show all work.

(1) <---2.1 2.2 2.3(open circle) 2.4 2.5--->
(2) <---2.1 2.2 2.3(filled circle) 2.4 2.5--->
(3) <---2.1 2.2 <---2.3(open circle) 2.4 2.5
(4) <---2.1 2.2 <---2.3(filled circle) 2.4 2.5

1. which inequality is represented by the graph below?

graph of a line y = 2x - 3 with shaded region

1. y ≤ 2x - 3
2. y ≥ 2x - 3
3. y ≤ -3x + 2
4. y ≥ -3x + 2

explain how you know using 2 distinct reasons:

1. lulani wants to buy a snowboard that has a total cost of $580, including tax. she has already saved $135 for it. at the end of each week, she is paid $96 for babysitting and is going to save three - quarters of that for the snowboard.

write an inequality that can be used to determine the minimum number of weeks lulani needs to babysit to have enough money to purchase the snowboard.
determine and state the minimum number of full weeks lulani needs to babysit to have enough money to purchase this snowboard.

Explanation:

Question 9

Step1: Solve the inequality \( 6.4 - 4x \geq -2.8 \)

Subtract \( 6.4 \) from both sides: \( -4x \geq -2.8 - 6.4 \)
Simplify: \( -4x \geq -9.2 \)
Divide both sides by \( -4 \) (reverse inequality): \( x \leq \frac{-9.2}{-4} = 2.3 \)

Step2: Analyze the number lines

• A closed dot (filled circle) means \( x \) includes \( 2.3 \).
• The arrow should point left (since \( x \leq 2.3 \)).

Check the options:

• (1): Open dot, right arrow → Incorrect.
• (2): Closed dot, right arrow → Incorrect (should point left).
• (3): Open dot, left arrow → Incorrect (needs closed dot).
• (4): Closed dot, left arrow → Correct.

1. Slope and Equation: The line has a slope of \( 2 \) (rise \( 2 \), run \( 1 \)) and y-intercept \( -3 \), so the line is \( y = 2x - 3 \).
2. Shading and Inequality Sign: The shading is above the line, so the inequality is \( y \geq 2x - 3 \) (dashed line? Wait, no—wait, the graph’s line: if it’s solid, but here the line is solid? Wait, the options: \( y \geq 2x - 3 \) (option 2) matches the line \( y = 2x - 3 \) and shading above (since “\( \geq \)” means above/below? Wait, for \( y \geq mx + b \), shading is above the line. The graph’s shading is above the line \( y = 2x - 3 \), so \( y \geq 2x - 3 \) (option 2) is correct.

Step1: Define Variables

Let \( w \) = number of weeks.

Step2: Calculate Weekly Savings

She saves \( \frac{3}{4} \) of \( \$96 \) per week: \( \frac{3}{4} \times 96 = 72 \) dollars per week.

Step3: Total Savings

Initial savings: \( \$135 \).
Total savings after \( w \) weeks: \( 135 + 72w \).

Step4: Inequality for “Enough Money”

Total savings must be at least \( \$580 \):
\( 135 + 72w \geq 580 \)

Answer:

(4)

Question 10