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Chapter 11: Problem 11

\(3-16=\) Graph the inequality. $$ 4 x+5 y<20 $$

Short Answer

Expert verified
Rearrange to slope-intercept form, plot line, and shade below.

Step by step solution

01

Rearrange the Inequality

We need to rearrange the inequality \(4x + 5y < 20\) into the slope-intercept form (\(y = mx + b\)) for easier graphing.Subtract \(4x\) from both sides:\[5y < -4x + 20\] Now, divide every term by 5:\[y < -\frac{4}{5}x + 4\].
02

Find the Y-intercept

The y-intercept (where the line crosses the y-axis) is found by setting \(x = 0\). From \(y < -\frac{4}{5}x + 4\), substituting \(x = 0\) gives \(y = 4\).So, the y-intercept is at the point \((0, 4)\).
03

Find the Slope

The slope \(m\) from the equation \(y < -\frac{4}{5}x + 4\) is \(-\frac{4}{5}\).This tells us that for every 5 units moved horizontally to the right, the line moves down 4 units vertically.
04

Plot the Line

Start plotting at the y-intercept \((0, 4)\).From \((0, 4)\), use the slope \(-\frac{4}{5}\) to find another point: move 5 units right (to \(x = 5\)) and 4 units down (to \(y = 0\)), marking the point \((5, 0)\).Draw a dashed line through these points since the inequality is '<', not '鈮�'.
05

Determine the Shaded Region

Since the inequality is \(y < -\frac{4}{5}x + 4\), the region below the dashed line should be shaded.Check by testing a point not on the line, such as \((0, 0)\). Substitute in the inequality:\[0 < -\frac{4}{5}(0) + 4\]\[0 < 4\] which is true, so shade this side of the line.

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Key Concepts

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

Slope-Intercept Form
Understanding the slope-intercept form is key when graphing linear equations and inequalities. This form is represented as \( y = mx + b \). Here, \( m \) is the slope of the line and \( b \) is the y-intercept. This format makes it easier to identify how a line behaves on a graph.
For instance, if we take the inequality \( 4x + 5y < 20 \), by rearranging it, we reach the slope-intercept form: \( y < -\frac{4}{5}x + 4 \). Notice how the left side is now simply \( y \), proceeding with the graph plotting becomes much simpler since you can see where the line will cross the y-axis and how it will slant due to the slope.
Y-Intercept
The y-intercept is a fundamental part of plotting any linear equation or inequality. It is where the line intersects the y-axis on a graph. In the slope-intercept form equation, \( y = mx + b \), the y-intercept is represented by \( b \).
Let's look at our example: \( y < -\frac{4}{5}x + 4 \). Here, the \( b \) value, or y-intercept, is 4. This means that the line crosses the y-axis at the point \( (0, 4) \). Identifying this point is the first step when you begin plotting the line on a graph because it provides a fixed starting point for you to apply the slope.
Graph Plotting
Graph plotting involves drawing the line that represents your equation or inequality on the coordinate plane. Before beginning, you should know the slope and y-intercept.
Start by plotting the y-intercept, in our inequality: \( (0, 4) \). Then, use the slope, which in this case is \(-\frac{4}{5}\). This tells us to go 5 units to the right horizontally and 4 units down vertically.
After identifying another point, such as moving to \( (5, 0) \) using the slope, you draw a line through these two points. For inequalities such as \( y < -\frac{4}{5}x + 4 \), use a dashed line to indicate that points on the line itself are not included in the solution set.
Inequality Shading
Once the line is plotted, shading the correct region is crucial to demonstrate the solution set of the inequality. Inequality shading shows where all possible solutions lie on a graph.
For \( y < -\frac{4}{5}x + 4 \), because the inequality is a "less than" sign, the area below the dashed line should be shaded. This means all y-values in the shaded region are smaller than the line values.
To confirm you're shading the correct region, choose a test point not on the line鈥攍ike \( (0, 0) \). Substitute this point back into the original inequality. If true, it confirms the area to shade. In our case, \( 0 < 4 \) is true, so shading below the line is correct.

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Most popular questions from this chapter

Nutrition A cat food manufacturer uses fish and beef byproducts. The fish contains 12 \(\mathrm{g}\) of protein and 3 \(\mathrm{g}\) of fat per ounce. The beef contains 6 \(\mathrm{g}\) of protein and 9 \(\mathrm{g}\) of \(\mathrm{fat}\) peounce. Fach can of cat food must contain at least 60 \(\mathrm{g}\) of protein and 45 \(\mathrm{g}\) of fat. Find a system of inequalities that describes the possible number of ounces of fish and beef that can be used in each can to satisfy these minimum requirements. Graph the solution set.

Use Cramer鈥檚 Rule to solve the system. $$ \left\\{\begin{aligned} \frac{1}{3} x-\frac{1}{5} y+\frac{1}{2} z &=\frac{7}{10} \\\\-\frac{2}{3} x+\frac{2}{5} y+\frac{3}{2} z &=\frac{11}{10} \\\ x-\frac{4}{5} y+z &=\frac{9}{5} \end{aligned}\right. $$

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don鈥檛 calculate the inverse. $$ \left[\begin{array}{rrrr}{1} & {3} & {3} & {0} \\ {0} & {2} & {0} & {1} \\\ {-1} & {0} & {0} & {2} \\ {1} & {6} & {4} & {1}\end{array}\right] $$

21-46 . Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$ \left\\{\begin{aligned} y & \geq x^{3} \\ y & \leq 2 x+4 \\ x+y & \geq 0 \end{aligned}\right. $$

Polynomials Determined by a Set of Points We all know that two points uniquely determine a line \(y=a x+b\) in the coordinate plane. Similarly, three points uniquely determine a quadratic (second-degree) polynomial $$ y=a x^{2}+b x+c $$ four points uniquely determine a cubic (third-degree) polynomial $$ y=a x^{3}+b x^{2}+c x+d $$ and so on. (Some exceptions to this rule are if the three points actually lie on a line, or the four points lie on a quadratic or line, and so on.) For the following set of five points, find the line that contains the first two points, the quadratic that contains the first three points, the cubic that contains the first four points, and the fourth-degree polynomial that contains all five points. $$ (0,0),(1,12), \quad(2,40), \quad(3,6), \quad(-1,-14) $$ Graph the points and functions in the same viewing rectangle using a graphing device.
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