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Chapter 9: Problem 4

Graph each inequality. $$x-y<6$$

Short Answer

Expert verified
The graph of the inequality \(x - y < 6\) (or equivalently \(y > x - 6\)) is a straight line with a slope of 1 and a y-intercept of -6, with all the region above the line shaded to represent the solution set of the inequality.

Step by step solution

01

Rearrange the inequality

First, rearrange the inequality so that it is in the form y = mx + b. This will make it easier to graph. The inequality \(x - y < 6\) can be rearranged to \(y > x - 6\).
02

Graph the line

Now graph the line based on the inequality you've rearranged. In this case, the line is y = x - 6. The y-intercept of the line is -6 and the slope is 1, meaning that for every unit increase in x, y increases by 1.
03

Shade the correct region

Finally, shading the region according to inequality. The inequality is \(y > x - 6\). This means y is greater than \(x - 6\), so you should shade the region above the line.

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Key Concepts

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

Linear Inequalities
A linear inequality is similar to a linear equation, but instead of using an equal sign, it uses inequality symbols like <, >, 鈮�, or 鈮�. In the context of two variables, such as our example "x - y < 6", it describes a region of the coordinate plane rather than a specific line. By rearranging the inequality into the form "y > x - 6", we can identify it as an inequality in the slope-intercept form, which is useful for graphing.
  • Linear inequalities have two variables: usually x and y.
  • They partition the plane into two regions鈥攐ne that satisfies the inequality and one that does not.
  • They can be strict (< or >) or inclusive (鈮� or 鈮�).
Understanding the nature of the inequality helps in determining how to shade the graph and where the solutions lie.
Graphing Techniques
Graphing an inequality uses techniques similar to those used in graphing linear equations, with some key differences. To graph the inequality \(x - y < 6\) effectively:
  • First, convert the inequality into equality, \(y = x - 6\), to initially draw the boundary line.
  • Determine whether the line should be solid or dashed. A dashed line is used for < or >, indicating the points on the line are not part of the solution.
  • Identify the slope and y-intercept from the rearranged form, \(y = x - 6\), where slope (m) is 1 and y-intercept (b) is -6.
After sketching the line, the next step is to determine which region to shade.
Coordinate Plane
The coordinate plane is essential for graphing equations and inequalities. It consists of two axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Every point in this plane is designated by an ordered pair \((x, y)\). When graphing inequalities:
  • Begin by plotting the y-intercept on the y-axis. In \(y > x - 6\), start at the point (0, -6).
  • Use the slope to find other points: the slope of 1 means you move up 1 unit for every 1 unit right.
  • Draw the boundary line and use it to help visualize the solution set.
Using the coordinate plane helps visualize which areas of the plane satisfy the inequality.
Inequality Solutions
Finding the solution to a linear inequality involves determining which region of the coordinate plane satisfies the inequality. For \(y > x - 6\), the solution is the area above the boundary line.
  • Select a test point not on the line, say (0,0), and substitute it into the inequality. \(0 > 0 - 6\) simplifies to \(0 > -6\), which is true.
  • If true, the region containing the test point is the solution set.
  • Shade this region to indicate the solutions.
These solutions make clear where any given x-y pair will satisfy the inequality, providing visual confirmation of the solution set.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I've noticed that when solving some compound inequalities with or, there is no way to express the solution set using a single interval, but this does not happen with and compound inequalities.

What does a dashed line mean in the graph of an inequality?

In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and \(2008 .\) Also shown is the percentage of households in which a person of faith is married to someone with no religion. (GRAPH CANT COPY) The formula $$1-\frac{1}{2} x+2=$$ models the percentage of U.S. households with an interfaith marriage, \(I, x\) years after \(1988 .\) The formula $$N=\frac{1}{4} x+6$$ models the percentage of U.S. households in which a person of faith is married to someone with no religion, \(N, x\) years after I988. Use these models to solve. a. In which years will more than \(33 \%\) of U.S. households have an interfaith marriage? b. In which years will more than \(14 \%\) of U.S. households have a person of faith married to someone with no religion? c. Based on your answers to parts (a) and (b), in which years will more than \(33 \%\) of households have an interfaith marriage and more than \(14 \%\) have a faith/no religion marriage? d. Based on your answers to parts (a) and (b), in which years will more than \(33 \%\) of households have an interfaith marriage or more than \(14 \%\) have a faith/no religion marriage?

What does a solid line mean in the graph of an inequality?

What does it mean if a system of linear inequalities has no solution?
See all solutions

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