Question

which statements are true about the graph of
y < \frac{2}{3}x + 1? check all that apply.
\square the slope of the line is 1.
\square the line is solid.
\square the area below the line is shaded.
\square a solution to the inequality is (2, 3).
\square the x-intercept of the boundary line is (-\frac{3}{2}, 0).
graph: y < \frac{2}{3}x + 1
click or tap the graph to plot a point.

Explanation:

Step1: Identify line slope

The inequality is in slope-intercept form $y=mx+b$, where $m$ is slope. Here $m=\frac{2}{3}$, so slope ≠1.

Step2: Check line type

Inequality uses $<$, not $\leq$, so line is dashed, not solid.

Step3: Determine shaded region

For $y < \text{linear expression}$, shade below the line.

Step4: Test point (2,3)

Substitute $x=2,y=3$ into inequality: $3 < \frac{2}{3}(2)+1 = \frac{4}{3}+1=\frac{7}{3}\approx2.33$. $3<2.33$ is false, so not a solution.

Step5: Find x-intercept

Set $y=0$ in boundary line $y=\frac{2}{3}x+1$:
$0=\frac{2}{3}x+1$
$\frac{2}{3}x=-1$
$x=-\frac{3}{2}$
So x-intercept is $(-\frac{3}{2},0)$.

Answer:

• The area below the line is shaded.
• The x-intercept of the boundary line is $(-\frac{3}{2}, 0)$.