10.6.2. Solving Systems of Linear Inequalities Graphically

Which of the points below belongs to the solution set of the system of inequalities represented by the figure shown?

A

B

C

D

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Example

Which of the points below belongs to the solution set of the system of inequalities represented by the figure shown?

A

B

C

D

Solution

The graph is in the first quadrant only, and the - and -axes are solid, so and . Also, the boundary lines of one of the
compound inequalities in the system are and . Since is solid, is dashed, and the region between them is coloured, the inequality
is . The boundary lines of the other compound inequality in the system are and . Since both lines are solid, and the region
between them is coloured, the inequality is . The boundary line of the additional inequality in the system passes through the points
and . Calculate the line's slope, or , as follows:

The line's -intercept, or , is 9, so in slope-intercept form, or , the line's equation is . The line is dashed, and the region
below it is coloured, so the inequality is . This means the system of inequalities is , , , , . Substitute the -coordinate of the point
into the inequalities and to get and . These are true statements. Substitute the -coordinate of the point into the
inequalities and to get and . These are also true statements. Now substitute the - and -coordinates of the point
into the inequality and simplify.

Since is a true statement as well, the point belongs to the solution set of the system of inequalities in the figure.