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Chapter 7: Problem 41

Graph each inequality. SEE EXAMPLE 4. (OBJECTIVE 2) $$x-2 y \leq 4$$

Short Answer

Expert verified
The graph of the inequality \(x-2y \leq 4\) is a straight, solid line starting at -2 on the y-axis, moving upward to the right with a slope of 1/2. The area above and including the line is shaded, denoting the solution to the inequality.

Step by step solution

01

Transpose the Inequality into Slope-Intercept Form

Rewrite the inequality in the form of y = mx + b. Start by isolating y in the inequality. To do this, we subtract x from both sides, resulting in \(-2y \leq 4 - x\). Divide all terms by -2 to solve for y. Because we are dividing by a negative number, the inequality sign flips, resulting in \(y \geq x/2 - 2\).
02

Graph the Boundary Line

Now we need to graph the equality \(y = x/2 - 2\), which will serve as the boundary line to indicate where the inequality either holds or doesn't hold. The y-intercept is -2 (where the line crosses the y-axis) and the slope is 1/2 (how steep the line is). Begin at -2 on the y-axis and use the slope, up 1 unit (rise) and right 2 units (run), to plot another point. Draw a straight line through these points. Because the inequality includes 'greater than or equal to', we will use a solid line to represent that points on the line are included in the solution.
03

Shade the Correct Side of the Boundary Line

Our next step is to shade the region of the graph that contains the solutions to the inequality. Since our inequality is \(y \geq x/2 - 2\), we need to shade the area above the line to denote all the points where y is greater than or equal to \(x/2 - 2\). If the inequality was \(y \< x/2 - 2\), we would have shaded the area below 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 graphing linear inequalities, the slope-intercept form is a very useful tool. It is written as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept. This form makes it easy to identify how steep the line is and where it crosses the y-axis.
In the initial exercise, the inequality \(x - 2y \leq 4\) was transformed into \(y \geq \frac{x}{2} - 2\) by isolating \(y\). Here, the slope \(m\) is \(\frac{1}{2}\) and the y-intercept \(b\) is \(-2\).
  • **Slope (\(m\))**: Represents the steepness of the line. A slope of \(\frac{1}{2}\) indicates that for every 2 units moved horizontally, the line moves 1 unit vertically.
  • **Y-intercept (\(b\))**: Indicates the point where the line crosses the y-axis, \((0, -2)\) in this case.
The simplicity of the slope-intercept form allows us to quickly sketch the boundary line of a graph by plotting the y-intercept and following the slope.
Boundary Line
The boundary line in inequality graphs is the line that illustrates where the solutions to the inequality begin or end. It's derived from converting the inequality to an equality.
For the inequality \(y \geq \frac{x}{2} - 2\), the boundary line is \(y = \frac{x}{2} - 2\). The key steps to plotting this line include:
  • **Identify the y-intercept**: Start at \(-2\) on the y-axis.
  • **Use the slope to find another point**: With a slope of \(\frac{1}{2}\), move up 1 unit and right 2 units from the y-intercept.
  • **Draw the line**: Connect the points with a straight line. Here, use a solid line because the inequality is 'greater than or equal to', meaning points on the line are included in the solution.
The boundary line is essential as it separates the plane into regions, one of which contains all the solutions to the inequality.
Shading Regions
Shading the correct region of the graph is crucial for indicating which set of points satisfies the inequality. Once the boundary line is graphed, the next step is to determine which side of the line to shade.
For an inequality like \(y \geq \frac{x}{2} - 2\), you shade above the boundary line. This represents that for any point in this region, \(y\) is greater than or equal to \(\frac{x}{2} - 2\).
Here's how you decide where to shade:
  • **Test a point**: Choose a point not on the line, like \((0, 0)\), and check if it satisfies the inequality.
  • **If true**: Shade the side of the line where the point lies.
  • **If false**: Shade the opposite side.
Shading visually demonstrates the solution set of the inequality, making it clear which regions include possible answers. Shifting the shade based on the inequality sign ensures clarity in problem-solving.

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Most popular questions from this chapter

Graph the solution set of each system of inequalities, when possible. SEE EXAMPLE 7. (OBJECTIVE 4) $$\left\\{\begin{array}{l} x \geq-1 \\ y>-2 \end{array}\right.$$

Graph the solution. $$\left\\{\begin{array}{l}\frac{x}{2}+\frac{y}{3} \geq 2 \\\\\frac{x}{2}-\frac{y}{2}<-1\end{array}\right.$$

Graph the solution. $$\left\\{\begin{array}{l}2 x-3 y<0 \\\2 x+3 y \geq 12\end{array}\right.$$

Determine whether each ordered pair is a solution of the given inequality. SEE EXAMPLE 1. (OBJECTIVE 1) Determine whether each ordered pair is a solution of \(x+y>4\) a. (0,4) b. (1,5) c. \(\left(-1, \frac{1}{2}\right)\) d. \(\left(-\frac{3}{4}, 7\right)\)

Graph the solution set of each system of inequalities, when possible. If not possible, state \(0 .\) SEE EXAMPLE 8. (OBJECTIVE 4) $$\left\\{\begin{array}{l}3 x+y<-2 \\\y>3(1-x)\end{array}\right.$$
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