Question

Which graph represents the solution set to the following system of linear inequalities?

First graph y-intercepts. second, use slope. third, determine if solid/dashed line.
y_>2x+7
y>-1/3x-2

The answer is B but can someone explain with the step by step notes mentioned?

Answer 1

B

Explanation:

Let's start by evaluating each of the equations:

`y\geq 2x+7`

To begin to understand this equation, first, graph line by replacing the inequality symbol with an equal sign:

The graph of this line is the first attached image at the bottom of this answer.

The inequality reads, y is greater than or equal to 2x + 7.

The "equal to" is very important. This means that the line
`y= 2x+7` is a solution to this problem, so the line is drawn as a SOLID line.

Because the inequality is GREATER than, that means that all values greater than the function are solutions! Thus the area ABOVE the line is shaded. The second image attached below shows the shaded graph.

Now, let's look at the second equation:

`y > -(1)/(3) x-2`

Start by replacing the inequality symbol with an equal sign and graphing the line. See the third image below for that line.

Now, THIS inequality is just GREATER than, not equal to. This means that the line that we just graphed,
`y = -(1)/(3) x-2` is NOT a solution to the equation! Therefore, the line is drawn as a dashed line. This means it is NOT included in the solution.

Again, since it is greater than, the area ABOVE the line is shaded. See the fourth image attached below to see the graph.

Now, let's combine these graphs! In the fifth image below, the blue represents
`y\geq 2x+7`, and the red represents
`y > -(1)/(3) x-2`.

The area where the red and blue overlap is the area to keep for the solution of this problem, since these equations represent a system of linear inequalities. I hope this helped!