/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 The graphs of solution sets of s... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 44

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises 43-44, you will be graphing the union of the solution sets of two inequalities. Graph the union of \(x-y \geq-1\) and \(5 x-2 y \leq 10\).

Short Answer

Expert verified
The union of the given inequalities covers all points on and below the line \(y = x + 1\) and on and above the line \(y = \frac{5}{2}x - 5\). The full region belongs to the union, and is the solution set of the inequalities.

Step by step solution

01

Graph the first inequality

The first inequality is \(x-y \geq-1\). Rearrange this equation to the y=mx+b form: \(y \leq x+1\). This line has a slope of 1 and y-intercept at 1. Graph this line on the coordinate plane. Shade the region below the line since this is a 'less than or equal to' inequality.
02

Graph the second inequality

The second inequality is \(5x-2y \leq 10\). Rearrange this equation to the y=mx+b form: \(y \geq \frac{5}{2}x-5\). This line has a slope of 5/2 and y-intercept at -5. Graph this line on the same coordinate plane. Shade the region above the line since this is a 'greater than or equal to' inequality.
03

Find the Union of both inequalities

The union of these two inequalities is the area covered by either of the shaded regions. It is not necessary for a point to satisfy both inequalities; if it satisfies either of them, it belongs to the union. The union, therefore, will be the entire shaded region in the graph.

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The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises 43-44, you will be graphing the union of the solution sets of two inequalities. Graph the union of \(y>\frac{3}{2} x-2\) and \(y<4\).

In Exercises 9-14, a. Determine if the parabola whose equation is given opens upward or downward. b. Find the vertex. c. Find the \(x\)-intercepts. d. Find the \(y\)-intercept. e. Use (a)-(d) to graph the quadratic function. \(y=x^{2}+8 x+7\)
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