Solving Systems of Inequalities

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Inequalities consist of variables and constants and they are arranged together separated by inequality signs like the greater than ‘>’ sign, lesser than sign ‘<’, greater than or equal to sign ‘≥’, and lesser than or equal to sign ‘≤’. If there is more than one inequation given, then it known as the system of inequalities and in order to solve this system, we have to graph each given inequality and have to look at their common region to get the solution set.

Example 1: Solve the given system of inequalities: x + y ≤ 1 and x – y ≤ 3.

Graph the inequality, x + y ≤ 1 treating it like a general equation.

Similarly graph the inequality x - y ≤ 3.

Now, shade the region of the given inequalities

according to their signs.

The red line represents x + y ≤ 1.

The green line represents x – y ≤ 3.

The common shaded region is the solution of the given system.

Example 2: Solve the given system of inequalities: x – y ≥ -1 and x + y ≤ 2.

Graph the inequality, x - y ≥ -1 treating it like a general equation.

Similarly graph the inequality x + y ≤ 2.

Now, shade the region of the given inequalities

according to their signs
.
The green line represents x – y ≥ -1.

The red line represents x + y ≤ 2.

The common shaded region is the solution of the given system.