# How do you graph the inequality y>=3/2x-3?

Jul 24, 2018

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#### Explanation:

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We have the inequality:

color(red)(y>=(3/2)x-3

If we can find both the x-intercept and the y-intercept, we can graph.

color(green)(Step.1

To find the x-intercept, set color(blue)(y=0

Replace color(red)(>= sign to color(red)(= for this step.

$y = \left(\frac{3}{2}\right) x - 3$

$0 = \left(\frac{3}{2}\right) x - 3$

Add color(red)(3 to both sides of the equality

$\Rightarrow 0 + 3 = \left(\frac{3}{2}\right) x - 3 + 3$

$\Rightarrow 3 = \left(\frac{3}{2}\right) x - \cancel{3} + \cancel{3}$

$\Rightarrow 3 = \left(\frac{3}{2}\right) x$

Divide both sides by color(red)(3/2

$\Rightarrow x = 3 \cdot \left(\frac{2}{3}\right)$

$\Rightarrow x = \cancel{3} \cdot \left(\frac{2}{\cancel{3}}\right)$

color(blue)( :. x= 2

Hence, x-intercept: color(blue)((2,0) ... Res.1

color(green)(Step.2

To find the y-intercept, set color(blue)(x=0

Replace color(red)(>= sign to color(red)(= for this step.

$y = \left(\frac{3}{2}\right) x - 3$

$y = \left(\frac{3}{2}\right) \left(0\right) - 3$

color(blue)( :. y= -3

Hence, y-intercept: color(blue)((0,-3) ... Res.2

Use both the intermediate results, Res.1 and Res.2, and plot the points on a graph.

Join the two points color(blue)(("x-intercept" and "y-intercept)" with a Solid Line, as our inequality uses a $=$ sign as well.

That would mean the value is a part of the solution.

color(green)(Step.3

Shading the Solution Region can be done as follows:

Use a Test Value to determine which part of the graph to shade.

Consider the point color(red)((0,0)

Substitute these values of color(blue)(x and y and test the inequality.

We have the inequality:

color(red)(y>=(3/2)x-3

$\Rightarrow 0 \ge \left(\frac{3}{2}\right) \cdot \left(0\right) - 3$

$0 \ge - 3$

This result is color(red)("TRUE"

So, the solution region is above the line of the graph.

And hence, our inequality graph will be: Solid line used indicates that the solution contains the values on the line.

Hope it helps.