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Chapter 9: Problem 33

Graph using either the test point or slope-intercept method. \(9 x-3 y \leq 21\)

Short Answer

Expert verified
The inequality \(9x - 3y \leq 21\) can be rewritten in slope-intercept form as \(y \geq -3x + 7\). To graph it, plot the y-intercept (0, 7) and use the slope (-3) to find another point on the line, such as (1, 4). Draw a solid line through these points, and shade the region above the line, where y values are greater than or equal to -3x + 7.

Step by step solution

01

Rewrite the inequality in slope-intercept form (y = mx + b)

We need to solve the inequality for y. Start with the given inequality: \(9x - 3y \leq 21\) Divide both sides by -3 to isolate y: \(y \geq -3x + 7\) Now, the inequality is in slope-intercept form, with m = -3 and b = 7.
02

Graph the line y = -3x + 7

Plot the y-intercept (0, 7) on the graph. Since the slope is -3, it means that for every 1 unit increase in x, the value of y decreases by 3 units. So from the y-intercept, move one unit to the right and three units down to find another point on the line, such as (1, 4). Then draw a line through these points.
03

Determine the shading region

Now we need to find which region the inequality represents, the one above or below the line. Since the inequality is in the form \(y \geq -3x + 7\), we should shade the region above the line, where y values are greater than or equal to the line.
04

Show the line as solid or dashed

To finish the graph, we need to show whether the line should be solid or dashed. Since the inequality states that \(y \geq -3x + 7\), which includes "greater than or equal to," the line will be solid, as it includes points on the line as valid solutions. The graph of the inequality \(9x - 3y \leq 21\) will have a solid line passing through points (0, 7) and (1, 4), with shading above 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
When working with inequalities or equations, using the slope-intercept form is incredibly helpful. The slope-intercept form is expressed as \(y = mx + b\), where \(m\) stands for the slope and \(b\) is the y-intercept.
Essentially, it tells us how steep the line is and where it intersects the y-axis.
For the inequality provided, \(9x - 3y \leq 21\), we first rearrange it into this form to make graphing easier. Here's how:
  • First, isolate \(y\) by adjusting both sides of the inequality until you have \(y\) by itself.
  • In the step provided, dividing both sides of the inequality by \(-3\) gives us \(y \geq -3x + 7\).
Now, it matches the slope-intercept form, where the slope \(m = -3\) and the y-intercept \(b = 7\). This helps us in plotting the line accurately on a graph. With the slope of \(-3\), the line declines quickly as \(x\) increases, which indicates a steeper descent.
Shading Regions
When you're graphing an inequality, the line divides the plane into two distinct regions. These regions need to be identified to know where the solutions to the inequality are located.
For the inequality \(y \geq -3x + 7\), the solution involves shading one of these regions.
  • The sign \(\geq\) tells us that we're looking for values above the line, where \(y\) is greater than or equal to the line's equation.
To do this practically:
  • First, draw the line using your slope and y-intercept.
  • Next, consider the orientation of the inequality. Since it shows \(y\) is greater, you'll shade the area above this line.
  • This visual distinction helps in clearly showing which side contains the solutions.
Remember, shading provides a quick visual check of where all the possible solutions lie.
Solid and Dashed Lines
When graphing inequalities, it's crucial to differentiate whether to use a solid or dashed line. This choice depends on whether the actual line is included in the solution to the inequality.
  • A solid line is used when the inequality sign includes equality, such as \(\geq\) or \(\leq\). This means points on the line are part of the solution.
  • A dashed line indicates that the points on the line are not part of the solution when the inequality is strict, like \(>\) or \(<\).
In our example, \(y \geq -3x + 7\), we use a solid line because the inequality includes equality (the \(\geq\) part), meaning every point on the line itself satisfies the inequality.
Thus, to accurately depict all solutions, both those on and above the line, a solid line is drawn through points such as \((0, 7)\) and \((1, 4)\). This attention to detail ensures that the graphed solution accurately represents all valid possibilities.

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Most popular questions from this chapter

Solve each system using Gaussian elimination. $$\begin{aligned}x-2 y+2 z &=3 \\\2 x-3 y+z &=13 \\\\-4 x-5 y-6 z &=8\end{aligned}$$

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The following exercises contain absolute value equations, linear inequalities, and both types of absolute value inequalities. Solve each. Write the solution set for equations in set notation and use interval notation for inequalities. $$\left|\frac{4}{3} x+1\right|=\left|\frac{5}{3} x+8\right|$$

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