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Chapter 3: Problem 6

Graph the inequality. $$ y<4 $$

Short Answer

Expert verified
Draw a dashed line at y = 4 and shade below it.

Step by step solution

01

Draw the Boundary Line

First, draw the line representing the equation y = 4y = 4 on the graph. Since the inequality is strictly less than 4, this will be a dashed line to indicate that points on the line are not included in the solution.
02

Identify the Shading Area

Next, determine where to shade. For the inequality y < 4y < 4, shade below the dashed line y = 4y = 4. This shaded region represents all the points (y-values) that are less than 4.
03

Verify a Point

Pick a point in the shaded area, such as (0, 0). Substitute these values into the inequality y < 4y < 4 to confirm that it satisfies the inequality. Since 0 < 40 < 4, the point (0, 0)(0, 0 lies in the solution region.

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Key Concepts

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

boundary line
When graphing an inequality, the first thing you need to do is draw the boundary line. This line is based on the equation obtained by turning the inequality symbol into an equality symbol. In our case, the inequality is \( y < 4 \), so the boundary line is represented by \( y = 4 \).
Because our inequality uses '<' (less than) and not '\( \le \)' (less than or equal to), the points on the line \( y = 4 \) are not part of our solution. As a result, we draw a dashed line instead of a solid line. A dashed boundary helps to visually convey that points on the line do not satisfy our inequality but all points just below do.
shading area
After drawing the boundary line, the next step is to identify where to shade.
This shading indicates the area where all points satisfy the given inequality. For our inequality \( y < 4 \), the shading will be below the boundary line \( y = 4 \). This is because 'y' being less than 4 means all values of 'y' that are smaller than 4.
To shade the correct area, you can select any point not on the boundary line. For instance, a common point to check for shading is (0,0). However, in this case, you do not need to substitute any point because the inequality is simple: every point below the dashed line \( y = 4 \) will satisfy \( y < 4 \).
verifying points
Verification is an essential step to ensure the correct region is shaded.
You do this by selecting a point that lies within the shaded area and substituting its coordinates back into the original inequality. For example, the point (0,0) lies in our shaded region. By substituting \( x = 0 \) and \( y = 0 \) into the inequality \( y < 4 \), and since \( 0 < 4 \) is a true statement, we verify that (0, 0) is indeed within the solution set.
It's always a good practice to verify multiple points, especially if the inequality or graph is more complex. This helps to double-check your shaded area is correct.

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Most popular questions from this chapter

(a) find three solutions of the equation. (b) graph the equation. \(y=x\)

(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. \((-30,-40) ;(10,30)\)

Use the slope formula to find the slope of the line that passes through the points. \((4 \mathrm{hr}, \$ 36) ;(6 \mathrm{hr}, \$ 54)\)

Use a graphing calculator to graph each equation. Choose a window that shows the \(x\)-intercept and \(y\)-intercept. Sketch the graph; describe the window. \(y=2 x+5\)

Use the slope formula to find the slope of the line that passes through the points. ( \(3 \mathrm{hr}, \$ 45) ;(7 \mathrm{hr}, \$ 105)\)
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