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Chapter 7: Problem 24

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x-y \geq 4 \\ x+y \leq 6\end{array}\right.\)

Short Answer

Expert verified
The solution to the system of inequalities is the region that lies above the line \(y = x-4\) and below the line \(y = -x + 6\) on the coordinate plane.

Step by step solution

01

Rearrange inequalities

Rewrite both inequalities in slope-intercept form (y = mx + b). This gives \(y \leq -x + 6\) and \(y \geq x - 4\).
02

Graph the inequalities

These can now be graphed like any other line. Remember that \(y \leq -x + 6\) is a downwards sloping line (because m=-1) crossing the y-axis at +6, and \(y \geq x - 4\) is an upwards sloping line (because m=1) crossing the y-axis at -4. For \(y \leq -x + 6\), because of the \'\(\leq\)\' inequality, all the points below the line are the possible y values (shade below the line). For \(y \geq x - 4\), because of the \'\(\geq\)\' inequality, all the points above the line are the possible y values (shade above the line).
03

Find the overlap

The solution to the system of inequalities is where the shaded regions of the individual inequalities overlap. This is the region that satisfies both inequalities. On the graph, this is where the upper region of the upward-sloping line overlaps with the lower region of the downward-sloping line.

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Most popular questions from this chapter

Compare the graphs of \(3 x-2 y>6\) and \(3 x-2 y \leq 6\). Discuss similarities and differences between the graphs.

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}y>2 x-3 \\ y<-x+6\end{array}\right.\)

Describe the shape of a scatter plot that suggests modeling the data with an exponential function.

Graph each linear inequality. \(y>-4\)

Graph each linear inequality. \(y \geq 0\)
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