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Solving Systems of Linear Inequalities Graphically

Solving Systems of Linear Inequalities Graphically 1.9 - Solution

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Let be the number of hours Neyma works per week as a painter and the number of hours she works per week as a waitress. Since Neyma makes per hour working as a painter and per hour in the second job, she will earn per week. Since she needs to earn at least this expression has to be greater than or equal to We also know that Neyma doesn't want to work more than hours per week, so we can write the second inequality. The system of inequalities is then

First we isolate on one of the sides to graph the inequalities.
We will graph and These are linear functions written in slope-intercept form so we can immediately identify the -intercepts as and and the slopes as and .

Now we have to determine if the solution sets lie above or below the lines. The first inequality is and describes all -coordinates larger than or equal to . This means that we have to shade the area above the solid blue line.

The graph of the second inequality

shows a solid green line with shading below it.

The solution to this system of inequalities is area where the shadings overlap. We distinguish this area in the graph below.


To name two possible solution, we need any two points from the overlapping area. We will plot the points in the graph and name them.

Two possible solutions are and