Question

Write the inequality that is graphed below.

Answer 1

Answer:
`y \ge 2x-7`

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Step-by-step explanation:

We first need to determine the equation of the boundary line.

That boundary goes through (0,-7) and (1,-5)

Let's compute the slope.

`(x_1,y_1) = (0,-7) \text{ and } (x_2,y_2) = (1,-5)\\\\m = \text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}}\\\\m = \frac{\text{y}_(2) - \text{y}_(1)}{\text{x}_(2) - \text{x}_(1)}\\\\m = (-5 - (-7))/(1 - 0)\\\\m = (-5 + 7)/(1 - 0)\\\\m = (2)/(1)\\\\m = 2\\\\`

The transition from (0,-7) to (1,-5) is "up 2, right 1"

• slope = rise/run = 2/1
• rise = 2, run = 1

The y intercept of the boundary line is b = -7 as this is where the boundary line crosses the y-axis.

We have m = 2 and b = -7; which means we go from y = mx+b to y = 2x-7

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The key takeaway of that previous section is that y = 2x-7 is the equation of the boundary line.

We then shade above the boundary. We'll replace the "equals" sign with a "greater than or equal to" sign.

Therefore, the final answer is
`y \ge 2x-7`

Take notice that the boundary line is solid. Therefore, points on the boundary are part of the shaded solution set. This is why "or equal to" is part of the inequality sign.

If the boundary was a dashed line, then we wouldn't have the "or equal to".

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Check:

The origin (0,0) is in the red shaded region. Plug those coordinates into the answer we found earlier.

`y \ge 2x-7\\\\0 \ge 2*0-7\\\\0 \ge 0-7\\\\0 \ge -7\\\\`

This is a true statement, so
`y \ge 2x-7` is true when (x,y) = (0,0). It confirms that (0,0) is in the shaded region. I'll let you try other points in the shaded region. Also, for more practice, try points that are on the boundary and points not in the shaded region.

GeoGebra and Desmos are great tools for quickly verifying the answer.