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Chapter 7: Problem 30

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x \geq 4 \\ y \leq 2\end{array}\right.\)

Short Answer

Expert verified
The solution set of the system of inequalities is the region on the coordinate plane where \(x\) is greater than or equal to 4 and \(y\) is less than or equal to 2. The region is represented by the area to the right of the line \(x = 4\) and below the line \(y = 2\). On a graph, it corresponds to the area in the lower right quadrant of the intersection of these two lines.

Step by step solution

01

Draw the Inequality \(x \geq 4\)

First, draw the inequality \(x \geq 4\) on a coordinate plane. Start with a vertical line at \(x = 4\). As the inequality allows \(x\) to be greater than or equal to 4, fill in the region to the right of this line or, equivalently, shade the region to the right.
02

Draw the Inequality \(y \leq 2\)

Then, graph the inequality \(y \leq 2\). Begin with a horizontal line at \(y = 2\). As \(y\) can be less than or equal to 2, the region to fill in or shade for this inequality is below the line.
03

Intersection of the Inequalities

Last but not least, find the intersection of these two areas. The solution area of a system of inequalities is the overlap of the individual solution areas of each inequality. As such, the intersection of the shaded regions from the first two steps gives the solution set for the system of inequalities.

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Most popular questions from this chapter

Graph each linear inequality. \(2 y-3 x>6\)

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 4 \\ \hline 1 & 5 \\ \hline 2 & 7 \\ \hline 3 & 11 \\ \hline 4 & 19 \\ \hline \end{array} $$

Without graphing, Determine if each system has no solution or infinitely many solutions. \(\left\\{\begin{array}{l}3 x+y \leq 9 \\ 3 x+y \geq 9\end{array}\right.\)

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

In Exercises 1-22, graph each linear inequality. \(x+y \geq 2\)
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