how do you solve and graph variable inequalities?
http://www.purplemath.com/modules/syslneq.htm
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To solve and graph variable inequalities, follow these steps:
1. Start by treating the inequality sign (>, <, ≥, or ≤) as an equal sign (=).
2. Solve the resulting equation to find the values of the variable that make the equation true.
3. If there is a variable (such as x), identify the critical values by setting the expression on one side of the equation to zero. These critical values are the points where the inequality may change.
4. Plot the critical values as open circles (for < and >) or closed circles (for ≤ and ≥) on a number line.
5. Based on the values of the variable, choose one test point from each interval between the critical values.
6. Substitute these test points into the original inequality. If the inequality is true, shade the interval containing that test point. If it is false, shade the other interval.
7. Finally, draw an arrow towards the shaded region to indicate the solution set.
For graphing linear inequalities with two variables, follow these steps:
1. Start by rearranging the inequality to the form y = mx + b, where m is the slope and b is the y-intercept.
2. Graph the line using the y-intercept (b) as a reference point and the slope (m) to determine other points on the line. Use a straight line for < or > and a solid line for ≤ or ≥.
3. Choose a test point not on the line. Substitute the x and y-values of that point into the inequality.
4. If the inequality is true, shade the region containing the test point. If it is false, shade the other region.
5. Finally, label the shaded region and draw arrows to indicate the solution set.
Remember, the solution set for an inequality is represented graphically by shaded regions on a number line or coordinate plane.