Advanced Linear Inequalities

I should also remind you that there are four types of inequalities: $>$ (greater than), $\ge$ (greater than or equal to), $\le$ (less than or equal to), and $<$ (less than). The type of inequality we wish to graph will crucially change the graph, even if the linear portion is the same.

One last thing: we are going to be using the slope-intercept form in this explanation. When we graphed equations, they were in the $y=mx+b$ format; in inequalities, the $=$ sign will be replaced with a sign representing one of the four types of inequalities.

What do you do when you receive an inequality in slope-intercept form? Here's a 3-step manual on how to graph the inequality.

Step 1: Graph the line as if it were a linear equation, rather than an inequality.

Step 2: If your inequality is either $>$ or $<$, with NO EQUAL SIGN AT THE BOTTOM, make the line dotted. Otherwise, move on to Step 3.

Step 3: If your inequality is either $>$ or $\ge$, that is, some type of GREATER THAN inequality, shade above the line (in the direction of the positive y-axis. Otherwise, shade below the line (in the direction of the negative y-axis).

Step 4: Check. Plug in the coordinates of a point that is shaded in your graph. If the inequality holds, great work! Otherwise, go back to Step 1 and fix any error.

Example: Graph the inequality $y \le 4-2x$.

Step 1: Graph the line as if it were a linear equation, rather than an inequality.

Here's Step 1!

Step 2: If your inequality is either $>$ or $<$, with NO EQUAL SIGN AT THE BOTTOM, make the line dotted. Otherwise, move on to Step 3.

Since we have a $\le$ inequality, we move on to Step 3.

Step 3: If your inequality is either $>$ or $\ge$, that is, some type of GREATER THAN inequality, shade above the line (in the direction of the positive y-axis. Otherwise, shade below the line (in the direction of the negative y-axis).

Since we have a $\le$, some type of LESS THAN inequality, we shade below the line.

Step 4: Check. Plug in the coordinates of a point that is shaded in your graph. If the inequality holds, great work! Otherwise, go back to Step 1 and fix any error.

For the sake of simplicity, let's just check the shaded point $(0, 0)$:

$0 \le 4 - 2(0) = 4 --> 0 \le 4$. This is true.

Therefore, we graphed the inequality correctly :-)

Now go do the exercises to practice your skills. Best of luck!