# Section 4-4

# Inequalities and Absolute Value Equations

GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES

To graph a linear inequality in two variables (say, *x* and *y*), first get *y* alone on one side. Then consider the **related equation **obtained by changing the inequality sign to an equals sign. The graph of this equation is a line.

If the inequality is **strict** (< or >), graph a dashed line. If the inequality is **not strict** ( or ), graph a solid line.

Finally, pick one point not on the line ((0, 0) is usually the easiest) and decide whether these coordinates satisfy the inequality or not. If they do, shade the half-plane containing that point. If they don't, shade the other half-plane.

Example:

Graph the inequality **y**** 4***x* – 2.

This line is already in slope-intercept form, with *y* alone on the left side. Its slope is 4 and its *y*-intercept is –2. So it's straightforward to graph it. In this case, we make a **solid** line since we have a "less than or equal to" inequality.

Now, substitute *x* = 0, *y* = 0 to decide whether (0, 0) satisfies the inequality.

This is false. So, shade the half-plane which does **not** include the point (0, 0).

GRAPHING ABSOLUTE VALUE EQUATIONS

The function *y* = |*x*| looks like this:

In the equation *y* = |*kx*|, the constant *k* has the effect of shrinking or widening the "V".