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Chapter 7: Problem 35

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

Short Answer

Expert verified
The solution set of the system of inequalities is the highlighted region on the graph which represents all the pairs of coordinates that satisfy both inequalities simultaneously.

Step by step solution

01

Plotting the inequality \(y > 2x - 3\)

Start by graphing the line \(y = 2x - 3\). Draw this line on the graph as a dashed line because the inequality symbol is '>' and points on the line do not satisfy the inequality. Then choose a test point not on the line, like (0,0). Substituting 0 for x and y in the inequality \(y > 2x - 3\), it yield 0 > -3, which is true, hence shade the area above the line to indicate the solutions for the inequality.
02

Plotting the inequality \(y < -x + 6\)

Similarly, graph the line \(y = -x + 6\). This will also be a dashed line because of the '<' symbol in the inequality. Now choose a test point, for instance (0,0), and substitute into the inequality \(y < -x + 6\). It gives 0 < 6, which is a true statement. Therefore, shade area below the line to indicate solutions of the inequality.
03

Determination of the solution set

The solution set for the system of inequalities is the intersection of the shaded areas from Step 1 and Step 2. Highlight this region on your graph. This region represents all pairs of (x, y) that satisfy both inequalities simultaneously.

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Most popular questions from this chapter

Use a table of coordinates to graph each exponential function. Begin by selecting \(-2,-1,0,1\), and 2 for \(x\). \(y=2^{x+1}\)

Graph each linear inequality. \(y>\frac{1}{4} x\)

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. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 4 \\ \hline 1 & 5 \\ \hline 2 & 7 \\ \hline 3 & 11 \\ \hline 4 & 19 \\ \hline \end{array} $$

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. $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & 5 \\ \hline 1 & 3 \\ \hline 2 & 1 \\ \hline 3 & -1 \\ \hline 4 & -3 \\ \hline \end{array} $$

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