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# Solving Linear Inequalities

Solving a linear inequality means writing it in a form so that only y appears on the left-hand side, and only x and constants appear on the right side. This is similar to solving an equation for y. It is easy to graph the solution sets for linear inequalities. Simply graph the corresponding boundary line and then shade the region either above or below the line.

Solving linear inequalities can be done by using the Addition and Multiplication Properties of Inequalities, just as the corresponding properties of equations are used to solve linear equations. This is best illustrated by examples.

Example 1

Solve y + 7 < 2 x + 5.

Solution

First, rewrite the inequality so that y is on the left-hand side by itself.

 y + 7 < 2 x + 5 Subtract 7 from each side. y < 2 x - 2

So, the solution set is the set of all points that lie below the line y = 2x - 2, as shown in the graph below. Notice that there is a dashed line on the boundary, since the inequality symbol is < , not .

Example 2

Solve 2y x + 6.

Solution

To solve the inequality, first divide each side by 2 to isolate the y .

 2y x + 6 y  Divide each side by 2.

The solution set is the set of points on and above the line , as shown in the graph below Example 3

Solve 3 - y x + 5.

Solution

First, use the properties of inequalities to isolate the y.

 3 - y x + 5 - y x + 2 Subtract 3 from each side. y - x - 2 Multiply each side by -1. Reverse the inequality symbol.

The graph of y x - 2 is shown below. Notice that a solid line is on the boundary of the region.

Recall that the Multiplication Property of Inequalities states that multiplying each side of an inequality by the same positive number produces an equivalent inequality. However, when multiplying each side by the same negative number, the inequality symbol must be reversed to get an equivalent inequality.