## How to Graph Linear Inequalities?

Whenever you graph inequalities, you’ll graph the ordinary linear functions the same as we did before. But the way it differs is the answer for the inequality isn’t the drawn line, it’s the area of the coordinate plane which satisfies the inequality.

The linear inequality divides the coordinate plane into $$2$$ halves via a boundary line (the line corresponding to the function). One side of the boundary line encompasses all the solutions for the inequality.
The boundary line is dashed for $$>$$ and $$<$$ and solid for $$≥$$ and $$≤$$.

Example: $$y \ ≥ \ 2x \ - \ 2$$

In the figure you will notice one side is colored gray and the other white, in order to figure out which side represents $$y \ ≥ \ 2x \ - \ 2$$, test a point. You test the point $$(0,0)$$ which is on the gray side.
$$y \ ≥ \ 2x \ - \ 2 \ ⇒ \ 0 \ ≥ \ 2(0) \ − \ 2 \ ⇒ \ 0 ≥ \ -2$$
The gray side is the one symbolizing the inequality $$y \ ≥ \ 2x \ - \ 2$$.

### Summary

• Firstly, graph the “equals” line.
• Pick a test point. (any point on both sides of the line.)
• Place the value of $$(x,y)$$ of that point in the inequality. If it works, that part of the line is the answer. If the values do not work, then the other part of the line is the answer.

### Exercises for Graphing Linear Inequalities

1) Sketch the graph of the Inequality: $$-15y \ - \ 6x \ < \ 30$$

2) Sketch the graph of the Inequality: $$11y \ + \ 22x \ < \ 11$$

3) Sketch the graph of the Inequality: $$-2y \ > \ -6x \ + \ 8$$

4) Sketch the graph of the Inequality: $$3y \ - \ 3x \ > \ 15$$

5) Sketch the graph of the Inequality: $$-4y \ > \ -16x \ - \ 4$$

6) Sketch the graph of the line: $$-7y \ - \ 7x \ < \ 14$$

7) Sketch the graph of the line: $$\frac{1}{3} \ y \ - \ x \ < \ -\frac{2}{3}$$

8) Sketch the graph of the line: $$3y \ - \ 6x \ > \ 3$$

9) Sketch the graph of the line: $$-5y \ + \ \frac{5}{2} \ x \ > \ -15$$

10) Sketch the graph of the line: $$-6y \ - \ 4x \ > \ -12$$

1) Sketch the graph of the Inequality: $$-15y \ - \ 6x \ < \ 30$$

$$\color{red}{-15y \ - \ 6x \ < \ 30 \ ⇒ \ y \ > \ -\frac{2}{5} \ x \ - \ 2 \ ⇒}$$

2) Sketch the graph of the Inequality: $$11y \ + \ 22x \ < \ 11$$

$$\color{red}{11y \ + \ 22x \ < \ 11 \ ⇒ \ y \ < \ -2x \ + \ 1 \ ⇒}$$

3) Sketch the graph of the Inequality: $$-2y \ > \ -6x \ + \ 8$$

$$\color{red}{-2y \ > \ -6x \ + \ 8 \ ⇒ \ y \ < \ 3x \ - \ 4 \ ⇒}$$

4) Sketch the graph of the Inequality: $$3y \ - \ 3x \ > \ 15$$

$$\color{red}{3y \ - \ 3x \ > \ 15 \ ⇒ \ y \ > \ x \ + \ 5 \ ⇒}$$

5) Sketch the graph of the Inequality: $$-4y \ > \ -16x \ - \ 4$$

$$\color{red}{-4y \ > \ -16x \ - \ 4 \ ⇒ \ y \ < \ 4x \ + \ 1 \ ⇒}$$

6) Sketch the graph of the line: $$-7y \ - \ 7x \ < \ 14$$

$$\color{red}{-7y \ - \ 7x \ < \ 14 \ ⇒ \ y \ > \ -x \ - \ 2 \ ⇒}$$

7) Sketch the graph of the line: $$\frac{1}{3} \ y \ - \ x \ < \ -\frac{2}{3}$$

$$\color{red}{\frac{1}{3} \ y \ - \ x \ < \ -\frac{2}{3} \ ⇒ \ y \ < \ 3x \ - \ 2 \ ⇒}$$

8) Sketch the graph of the line: $$3y \ - \ 6x \ > \ 3$$

$$\color{red}{3y \ - \ 6x \ > \ 3 \ ⇒ \ y \ > \ 2x \ + \ 1 \ ⇒}$$

9) Sketch the graph of the line: $$-5y \ + \ \frac{5}{2} \ x \ > \ -15$$

$$\color{red}{-5y \ + \ \frac{5}{2} \ x \ > \ -15 \ ⇒ \ y \ < \ \frac{1}{2} \ x \ + \ 3 \ ⇒}$$

10) Sketch the graph of the line: $$-6y \ - \ 4x \ > \ -12$$

$$\color{red}{-6y \ - \ 4x \ > \ -12 \ ⇒ \ y \ < \ -\frac{2}{3} \ x \ + \ 2 \ ⇒}$$

## Graphing Linear Inequalities Quiz

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