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Chapter 4: Problem 23

In Exercises 21-28, sketch the graph of the linear inequality. $$ y \leq 2 x-1 $$

Short Answer

Expert verified
The graph for the given inequality is a solid line at \(y = 2x - 1\) with the area below it shaded to represent all the points that satisfy \(y \leq 2x - 1\).

Step by step solution

01

Sketch the Line

First, we sketch the line \(y = 2x - 1\). The slope of this line is 2 (the coefficient of x) and its y-intercept is -1 (the constant). On a graph, we can start by marking the y-intercept (0,-1). From this point, because the slope is 2, we go up 2 units and over 1 unit to the right to place our second point. We then draw the line through these two points.
02

Choose the region

The inequality 'y \(\leq\) 2x -1' tells us that the y-values that we are interested in are less than or equal to the ones on our line. On a graph, this region is below the drawn line. We shade this lower area to represent all those points that satisfy the inequality.
03

Include the line in the solution

Because our inequality includes an equal sign (\(\leq\)), we must also include the line itself in our solution. We can represent this by making our line solid to show that points on the line also satisfy our inequality.

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Key Concepts

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

Graphing Inequalities
Graphing inequalities is a useful method to visually represent solutions of an inequality on a coordinate plane. When working with linear inequalities, it involves graphing a linear equation and shading a particular region on one side of the line.
By shading one side of the line, we indicate all the potential solutions to the inequality. Each point in the shaded area is a pair of x and y values that make the inequality true.
  • First, graph the related linear equation.
  • Choose which side of the line represents the solution to the inequality.
  • If the inequality includes 鈮� or 鈮�, draw a solid line. If it uses < or >, draw a dashed line to exclude the line from the solution.
Slope-Intercept Form
The slope-intercept form is a way of writing the equation of a line so that you can easily identify the slope and the y-intercept. It is expressed as:\[y = mx + b\]Where:
  • \( m \) is the slope of the line. This tells you how steep the line is.
  • \( b \) is the y-intercept of the line, which is where the line crosses the y-axis.
The slope-intercept form is particularly useful for quickly sketching the graph of a line and understanding how changes in the equation affect the line's position and orientation on the graph. To graph a line using this form, start by plotting the y-intercept and then use the slope to find another point.
Shading Regions
Shading regions helps visualize which parts of the graph are included in the solution set of the inequality. After graphing the line:
  • If the inequality is "less than or equal to" (\( \leq \)), shade the region below the line.
  • If it is "greater than or equal to" (\( \geq \)), shade the region above the line.
When the inequality does not include equality (\( < \) or \( > \)), ensure the line is dashed. Shading is a powerful tool to better understand the extent of solutions visually, offering a clear distinction between what values satisfy the inequality and which do not.
To double-check the shaded region, pick a test point not on the line and substitute it into the inequality. If true, your shading is correct.
Y-Intercept
The y-intercept is a fundamental concept when dealing with linear equations and inequalities. It signifies the point where the line intersects the y-axis.

In the equation \( y = 2x - 1 \), the y-intercept is \( -1 \). This means that the line will cross the y-axis at the point \( (0, -1) \).
  • The y-intercept is the point where the value of x is zero.
  • You can find it directly as the constant in the slope-intercept form equation.
Understanding the y-intercept allows for a quick start when graphing, as it provides a solid starting point to project the rest of the line using the slope.

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Most popular questions from this chapter

In Exercises 71-76, write an equation of the line that passes through the points. $$ \left(\frac{5}{6},-2\right),\left(\frac{1}{6}, 4\right) $$

Write an equation of the line that passes through the point and is (a) parallel and (b) perpendicular to the given line. $$ \begin{aligned} &(2,1) \\ &x-y=3 \end{aligned} $$

Determine the slope of the line. $$ 5 x+4 y-8=0 $$

In Exercises 21-28, sketch the graph of the linear inequality. $$ x+y>0 $$

In Exercises \(13-16\), sketch the graph of the linear inequality. $$ y>-5 $$
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