Question

identifying an inequality and solution from its graph
which inequality is represented by the graph?
y < 2x + 3/4
which point is a solution to the inequality?
(-2, -4)
(-1, -2)
(1, 2)
(2, 5)

Explanation:

Step1: Verify the given inequality

First, confirm the inequality matches the graph. The dashed line has slope 2 and y-intercept $\frac{3}{4}=0.75$, so the boundary is $y=2x+\frac{3}{4}$. The shaded region is below the line, so the inequality is $y<2x+\frac{3}{4}$.

Step2: Test each point in the inequality

Test $(-2,-4)$:

Substitute $x=-2, y=-4$ into $y<2x+\frac{3}{4}$:
$$-4 < 2(-2) + \frac{3}{4}$$
$$-4 < -4 + 0.75$$
$$-4 < -3.25$$
This is true.

Test $(-1,-2)$:

Substitute $x=-1, y=-2$ into $y<2x+\frac{3}{4}$:
$$-2 < 2(-1) + \frac{3}{4}$$
$$-2 < -2 + 0.75$$
$$-2 < -1.25$$
This is true.

Test $(1,2)$:

Substitute $x=1, y=2$ into $y<2x+\frac{3}{4}$:
$$2 < 2(1) + \frac{3}{4}$$
$$2 < 2 + 0.75$$
$$2 < 2.75$$
This is true.

Test $(2,5)$:

Substitute $x=2, y=5$ into $y<2x+\frac{3}{4}$:
$$5 < 2(2) + \frac{3}{4}$$
$$5 < 4 + 0.75$$
$$5 < 4.75$$
This is false.

Step3: Identify valid solutions

Points that satisfy the inequality are valid solutions.

Answer:

The confirmed inequality is $y<2x+\frac{3}{4}$.
The solution points are:
(-2, -4)
(-1, -2)
(1, 2)