Question

match the inequality $y \leq 2x + 3$ with its graph.

which of the given graphs is the correct graph of the inequality $y \leq 2x + 3$?
\\(\bigcirc\\) a.
\\(\bigcirc\\) b.
\\(\bigcirc\\) c.
\\(\bigcirc\\) d.
graphs for options a, b, c, d are shown with coordinate grids and shaded regions, along with zoom and other icons

Explanation:

Step1: Analyze the inequality form

The inequality is \( y \leq 2x + 3 \). First, consider the boundary line \( y = 2x + 3 \). The slope \( m = 2 \) and the y - intercept \( b = 3 \). Since the inequality is \( \leq \), the boundary line should be solid (because the inequality includes equality).

Step2: Determine the region to shade

To find which side of the line to shade, we can test a point not on the line. Let's use the origin \((0,0)\). Substitute \( x = 0 \) and \( y = 0 \) into the inequality: \( 0\leq2(0)+3\), which simplifies to \( 0\leq3 \), a true statement. So we shade the region that includes the origin.

Now, let's analyze the options:

• For a line \( y = 2x + 3 \), when \( x = 0 \), \( y = 3 \), so the y - intercept is at \((0,3)\). When \( y = 0 \), \( 0=2x + 3\Rightarrow x=-\frac{3}{2}=- 1.5\), so the x - intercept is at \((-1.5,0)\).
• The line should be solid (since \( \leq \)). Now, check the shading: since \((0,0)\) satisfies the inequality, the shaded region should include the origin.

Looking at the options, we need to find the graph with a solid line (since \( \leq \)) and shading that includes the origin. Let's assume the graphs:

• Option A: If the line is solid and the shading includes the origin (since when we test \((0,0)\), it's in the shaded region for the correct graph), and the line has slope 2 and y - intercept 3.
• Option B: If the line is dashed, it's incorrect because our inequality is \( \leq \) (should be solid).
• Option C: If the shading is on the wrong side (not including the origin), incorrect.
• Option D: If the line is dashed or shading is wrong, incorrect.

Assuming from the general form, the correct graph should have a solid line \( y = 2x+3 \) and shading below (or including) the line, and including the origin. So the correct option is the one with solid line and shading that includes \((0,0)\). Let's say the correct option is (assuming the first correct - looking one with solid line and correct shading) A (but we need to check the actual graphs, but from the process, the key is solid line and shading including origin).

Answer:

(Assuming the correct graph is, for example, A. But based on the analysis, the correct graph should have a solid line \( y = 2x + 3 \) and shading that includes the origin. So if A has a solid line and shading including origin, then) A. (The actual correct option depends on the visual graphs, but the process is as above.)