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Chapter 3: Problem 17

Graph the inequality. $$ y<-\frac{9}{4} x+7 $$

Short Answer

Expert verified
Graph the line \( y = -\frac{9}{4}x + 7 \) with a dashed line and shade the region below it.

Step by step solution

01

Rewrite the inequality

Rewrite the given inequality clearly: \[ y < -\frac{9}{4}x + 7 \]
02

Identify the slope and y-intercept

Compare the inequality to the slope-intercept form \( y = mx + b \). Here, the slope \( m \) is \( -\frac{9}{4} \) and the y-intercept \( b \) is 7.
03

Graph the boundary line

Graph the boundary line \( y = -\frac{9}{4}x + 7 \). Start at the y-intercept (0, 7). From this point, use the slope to find another point: move down 9 units and right 4 units to reach the point (4, -2). Plot both points and draw a dashed line through them since the inequality is strictly '<'.
04

Shade the appropriate region

Since the inequality symbol is '<', shade the region below the dashed line.
05

Verify a point

Choose a test point that is not on the line (e.g., (0,0)). Substitute it into the inequality: \[ 0 < -\frac{9}{4}(0) + 7 \implies 0 < 7 \] This is true, confirming that the shading is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form
The slope-intercept form is a way to write the equation of a line. This form is very useful in graphing inequalities. The slope-intercept form looks like this: \(y = mx + b\). Here, \(m\) is the slope of the line and \(b\) is the y-intercept.Understanding the slope-intercept form can make it easier to graph lines and, in this case, inequalities. By identifying the slope \(m\) and y-intercept \(b\), you can plot the line on a graph quickly.
y-intercept
The y-intercept is the point where the line crosses the y-axis. For the inequality \(y < -\frac{9}{4}x + 7\), the y-intercept is 7. This means that the line crosses the y-axis at the point (0, 7). To find the y-intercept in any equation written in slope-intercept form, look for the constant term \(b\). Graphing starts at this point. For example, in our inequality, you start by marking (0, 7) on the graph.
boundary line
The boundary line helps us separate the regions of the graph. For \(y < -\frac{9}{4}x + 7\), the boundary line is \(y = -\frac{9}{4}x + 7\). To draw this line, start at the y-intercept (0, 7). Use the slope to find another point. Our slope \(-\frac{9}{4}\) means that for every 4 units you move right, you go down 9 units. Plot the next point at (4, -2) and draw a dashed line through both points.We use a dashed line because our inequality is strictly '<', meaning points on the line are not part of the solution.
shading region
Shading a region shows where the solutions to the inequality lie. For \(y < -\frac{9}{4}x + 7\), you shade the area below the boundary line.To determine where to shade, pick a test point not on the line, like (0, 0). Substitute this point into the inequality. If true, that region is the solution.\[ 0 < -\frac{9}{4}(0) + 7 \implies 0 < 7 \]This is true, so you shade below the boundary line.

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Most popular questions from this chapter

According to the Pew Research Center, the Millennial Generation in the United States includes about 50 million people born after 1980 . Find the number of this generation who say that "you can't be too careful" when dealing with people. Round to the nearest tenth of a million. Whether as a by-product of protective parents, the age of terrorism or a media culture that focuses on dangers, they [Millennials] cast a wary eye on human nature. Two-thirds say "you can't be too careful" when dealing with people. (Source: www.pewresearch.org/millennials, 2010)

Use the slope formula to find the slope of the line that passes through the points. \((-5,4) ;(-9,3)\)

Use the slope formula to find the slope of the line that passes through the points. \((-6,-3) ;(-9,-2)\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((1,5) ;(3,-3)\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(\frac{5}{8}, \frac{1}{6}\right) ;\left(\frac{3}{8}, \frac{5}{6}\right)\)
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