# Represent Inequalities as Regions

## Representing Linear Inequalities as Regions

On the right, we have represent the inequality y≥3x−4. The first step to representing a linear inequality as a region is to replace the inequality sign with an equal sign. In this example, we would get y=3x-4. Draw the line for this equation an a graph and make it a normal line if it is smaller/greater or equal to and a dashed line if it is just smaller or greater than. In this case the line is not dashed as y≥3x−4. Then you need to shade in the graph on one side of the line. To do this, take any point on the graph that is not on the line and replace y and x in the inequality with the co-ordinates of the point. It is usually easiest to use (0,0). If we were to use the origin, we would get 0≥3(0)-4 which is equivalent to 0≥-4. Now you simply need to see if this inequality is correct. Since 0 is greater than or equal to -4, the origin is in the region of this inequality. Consequently, you need to shade the other side of the line (we tend to always shade out the area that is not in the region). Then write an R on the side of the line that is not shaded to make clear that this side is the region of the inequality.
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Example 1 This region represents the inequality y<2x−5. We started off by drawing a dashed line of y=2x-5 (the line is dashed because the inequality contains a < sign and not a ≤ sign). Then We replaced x and y in the equation with (0,0). Since 0<-5 is not correct, the point (0,0) cannot be in the region. As a result, you must shade out the side where this point is and put an R on the other side to show that this is the region.
Example 2 This region represents the inequality y≥5x−1. The first step was to draw a line of the equation y=5x-1. This time the line is not dashed as the inequality contains a greater than or equal to sign. We again used the origin (0,0) and substituted it with x and y. We got 0≥−1. Since this inequality is correct, we shade the opposite side of the line and put an R onto the side with the origin to show that the region of this inequality is on this side of the line.
R
R
R

## Representing Quadratic Inequalities as Regions

Representing quadratic inequalities as regions is very similar to representing linear ones. On the graph on the right, we have the inequality y≥2x²−3x−2. The first step is to draw the parabola of the quadratic. The best way to do this is to replace y≥ with 0= and then solve the quadratic and sketch it. Depending on the inequality sign, you have to use a dashed or normal line.  Once you have drawn the parabola, use any point to check if it is in the region or not. We will use (0,0). 0≥2(0)²-3(0)-2 which is equal to 0≥-2. Since this is correct, the origin is point in the region of the inequality. The origin is inside the parabola so any point outside is not in the region and has to be shaded out. Again, you can put an R into region of the inequality. Sometimes, your teacher may ask you to shade in instead of shading out. Then you just have to remove the R and shade the correct region of the inequality.
R
Example 1 This region represents the inequality y parabola of y=x²-3x+2 with a dashed line since we have y < sign. The we checked if the origin is in the region of the inequality by substituting x and y by (0,0): 0<2. Since the inequality is correct and the origin is outside of the parabola, the area inside the parabola has to be shaded out. You can put an R in the region that is not shaded to show that this is the region of the inequality.
Example 2 This region represents the inequality y≥-2x²-x+2. The first thing we have to do is draw the parabola for y=2x²-x+2 using a normal line as we have the ≥ sign in the inequality. Then we replace x and y with the co-ordinates of the origin (0,0) to check if the origin is in the region: 0≥2. Since this inequality is incorrect, the origin cannot be in the region. The origin is inside the parabola. Consequently,  we have to shade out the interior of the parabola. We can put an R outside the parabola to show that this is the region of the inequality.
R
R

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