Question

to $y + 4 \leq 7$
a. $y \leq 3$
\

$$\begin{tikzpicture} \\draw-> (-6,0) -- (6,0); \\foreach \\x in {-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6} \\draw (\\x,0.1) -- (\\x,-0.1) nodebelow {\\x}; \\drawthick, blue (-6,0) -- (3,0); \\fillblue (3,0) circle (2pt); \\end{tikzpicture}$$

b. $y \geq 3$
\

$$\begin{tikzpicture} \\draw-> (-6,0) -- (6,0); \\foreach \\x in {-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6} \\draw (\\x,0.1) -- (\\x,-0.1) nodebelow {\\x}; \\drawthick, blue (3,0) -- (6,0); \\fillblue (3,0) circle (2pt); \\end{tikzpicture}$$

c. $y < 3$
\

$$\begin{tikzpicture} \\draw-> (-6,0) -- (6,0); \\foreach \\x in {-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6} \\draw (\\x,0.1) -- (\\x,-0.1) nodebelow {\\x}; \\drawthick, blue (-6,0) -- (3,0); \\fillwhite (3,0) circle (2pt); \\drawblue (3,0) circle (2pt); \\end{tikzpicture}$$

d. $y > 3$
\

$$\begin{tikzpicture} \\draw-> (-6,0) -- (6,0); \\foreach \\x in {-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6} \\draw (\\x,0.1) -- (\\x,-0.1) nodebelow {\\x}; \\drawthick, blue (3,0) -- (6,0); \\fillwhite (3,0) circle (2pt); \\drawblue (3,0) circle (2pt); \\end{tikzpicture}$$

Explanation:

Step1: Solve the inequality

Start with the inequality \( y + 4 \leq 7 \). Subtract 4 from both sides: \( y + 4 - 4 \leq 7 - 4 \), which simplifies to \( y \leq 3 \).

Step2: Analyze the number line graphs

• Option A: The graph has a closed dot at 3 (since \( \leq \) includes 3) and the line extends to the left (for values less than or equal to 3), which matches \( y \leq 3 \).
• Option B: The graph extends to the right, representing \( y \geq 3 \), which does not match.
• Option C: The graph has an open dot (representing \( < \)), but our solution is \( \leq \), so this is incorrect.
• Option D: The graph has an open dot and extends to the right, representing \( y > 3 \), which is incorrect.

Answer:

A. \( y \leq 3 \) with the graph showing a closed dot at 3 and an arrow to the left.