Question

identifying an inequality and solution from its graph
which inequality is represented by the graph?
which point is a solution to the inequality?

Explanation:

Part 1: Find the inequality represented by the graph

Step 1: Find the slope of the line

The two points on the line are \((0, 0.75)\) and \((2, 4.75)\). The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{4.75 - 0.75}{2 - 0}=\frac{4}{2} = 2\).

Step 2: Find the equation of the line

Using the slope - intercept form \(y=mx + b\), where \(b = 0.75\) (the y - intercept) and \(m = 2\), the equation of the line is \(y=2x + 0.75\).

Step 3: Determine the inequality symbol

The line is dashed (so the inequality is either \(y>2x + 0.75\) or \(y<2x + 0.75\)) and the shaded region is above the line. So the inequality is \(y>2x+0.75\) (or \(y - 2x>0.75\), etc.).

Part 2: Find a point that is a solution to the inequality

We can test a point in the shaded region. Let's take the point \((0,2)\) (which is in the shaded region).
Substitute \(x = 0\) and \(y = 2\) into the inequality \(y>2x + 0.75\).
Left - hand side: \(y=2\)
Right - hand side: \(2x+0.75=2(0)+0.75 = 0.75\)
Since \(2>0.75\), the point \((0,2)\) is a solution. (We can also test other points in the shaded region, for example, \((1,3)\): \(y = 3\), \(2x+0.75=2(1)+0.75 = 2.75\), and \(3>2.75\))

Answer:

for Part 1:
The inequality represented by the graph is \(y>2x + 0.75\) (or equivalent forms like \(y-2x>0.75\))