Problem 27 In Exercises 21-28, sketch the g... [FREE SOLUTION]

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Elementary Algebra Within Reach

Ron Larson, Kimberly Nolting

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 4: Problem 27

In Exercises 21-28, sketch the graph of the linear inequality. $$ y>-2 x+10 $$

Short Answer

Expert verified

The graph starts with a dashed line representing $y = -2x + 10$, descending from left to right starting at the y-intercept (10). The region above this line is shaded, representing the solution to the inequality $y > -2x + 10$.

Step by step solution

Analyze the inequality

The inequality given is $y > -2x + 10$. This means that the solution to this inequality lies above the line $y = -2x + 10$. This line will be a boundary line, and because the inequality symbol does not contain 'equal to', the line will be dashed.

Plot the line

Start by plotting the line $y = -2x + 10$. This can be done by finding the y-intercept and the slope of the line. The y-intercept is the value of 'y' when 'x' is zero, which in this case is 10. The slope is the coefficient of 'x', which is -2. In this case, slope is negative, which means the line will descend from left to right.

Shade the region

The inequality symbol '>' denotes that the solutions to this inequality are greater than the line $y = -2x + 10$. This means that the solution set lies above the line. So, the region above the line should be shaded.

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Key Concepts

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

Graphing Linear Inequalities

Graphing linear inequalities may sound complex, but it's a straightforward process once you break it down. The goal is to visually represent all the solutions that satisfy an inequality. Let's start with understanding the given inequality: $ y > -2x + 10 $. It tells us the solutions are in the area above a particular line.

First, you'll need to sketch the related boundary line, which we will discuss next.
After plotting the boundary line, notice the inequality symbol ('>') hints that any solution is above this line.
Finally, shade the region above the boundary line, because that's where the solutions to the inequality exist.

Using these strategies makes graphing linear inequalities much more manageable.

Boundary Lines

Boundary lines play a crucial role in graphing linear inequalities. They act as a divider on your graph between where solutions do and do not lie. How do you plot them? Let's find out.

For the inequality $ y > -2x + 10 $, the boundary line is exactly where equality holds: $ y = -2x + 10 $.
Because our inequality is 'greater than' and doesn't include 'equal to', the boundary line should be dashed. This indicates that the points on the line are not part of the solution set.

Knowing when to use a dashed line versus a solid one is essential for accurately representing inequalities.

Slope-Intercept Form

The slope-intercept form is a key concept in understanding and graphing linear equations. This form is given by the equation $ y = mx + b $, where:

$ m $ is the slope of the line.
$ b $ is the y-intercept, where the line crosses the y-axis.

For our boundary line $ y = -2x + 10 $:

The slope $ m $ is $-2$. A negative slope indicates that the line descends from left to right.
The y-intercept $ b $ is $ 10 $, meaning the line crosses the y-axis at $ y = 10 $.

By identifying the slope and y-intercept, you can confidently plot the line that serves as your inequality's boundary.

Solution Set

The solution set of a linear inequality encompasses all the points that satisfy the inequality. Understanding which region to shade on the graph is crucial. Remember:

For $ y > -2x + 10 $, the solution set is all the points above the boundary line.
Think of the inequality symbol ('>') as a hint that the region of interest is above the line. If it were '<', you'd be looking below.
Each point in the solution set, when plugged into $ y > -2x + 10 $, makes the inequality true.

Visualizing the solution set helps you understand the range of values that satisfy the inequality and aids in solving more complex math problems.

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91影视

In Exercises 21-28, sketch the graph of the linear inequality. $$ y>-2 x+10 $$

Short Answer

Step by step solution

Analyze the inequality

Plot the line

Shade the region

Key Concepts

Graphing Linear Inequalities

Boundary Lines

Slope-Intercept Form

Solution Set

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