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Chapter 7: Problem 37

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x+2 y \leq 4 \\ y \geq x-3\end{array}\right.\)

Short Answer

Expert verified
The solution set for the system of inequalities is the shared shaded area above the line \(y = x - 3\) and below the line \(y = -\frac{1}{2}x + 2\).

Step by step solution

01

Transform Inequalities to Slope-Intercept Form

The given system of inequalities is \(\left\{\begin{array}{l}x+2 y \leq 4 \ y \geq x-3\end{array}\right.\). They can be rewritten in slope-intercept form \(y = mx + b\) by isolating \(y\). Inequality 1 becomes \(y \leq -\frac{1}{2}x + 2\) and Inequality 2 becomes \(y \geq x - 3\)
02

Graphing the Linear Inequalities

Start by graphing the boundary lines. For Inequality 1, the boundary is the line \(y = -\frac{1}{2}x + 2\) with a solid line because the inequality \(y \leq -\frac{1}{2}x + 2\) includes equal to portion. The line will have a slope of -1/2 and y-intercept of 2. Similarly, for Inequality 2, the boundary is the line \(y = x - 3\) with a solid line because the inequality \(y \geq x - 3\) includes equal to portion. It will have a slope of 1 and y-intercept of -3.
03

Shading the Regions of the Graph

Inequality 1 states that \(y\) is less than or equal to \(-\frac{1}{2}x + 2\). So, you should shade below this line. Inequality 2 suggests that \(y\) is greater than or equal to \(x - 3\). You should shade above this line. The shared shaded area between these two inequalities constitutes the solution for this system of inequalities.

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Most popular questions from this chapter

Graph each linear inequality. \(x \leq 2\)

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$ f(x)=62+35 \log (x-4), $$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the function to solve Exercises 37-38. a. According to the model, what percentage of her adult height has a girl attained at age ten? Use a calculator with a LOG key and round to the nearest tenth of a percent. b. Why was a logarithmic function used to model the percentage of adult height attained by a girl from ages 5 to 15 , inclusive?

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 each linear inequality. \(2 x-5 y<10\)

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{r}2 x+y<4 \\ x-y>4\end{array}\right.\)
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