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Chapter 7: Problem 29

Graph \(|x+y|<1\).

Short Answer

Expert verified
It's the region between the lines \(x+y=-1\) and \(x+y=1\) (dashed).

Step by step solution

01

Rewrite the Inequality

The inequality \(|x+y|<1\) can be rewritten in the form of a double inequality: \(-1 < x+y < 1\). This means we need to consider two separate inequalities: 1. \(x+y > -1\)2. \(x+y < 1\).
02

Graph the Boundary Lines

For each inequality, first graph the boundary line:1. For \(x+y=-1\): This line can be written as \(y = -x-1\). Graph this line on the coordinate plane. Use a dashed line to indicate that points on this line are not included in the solution.2. For \(x+y=1\): This line can be rewritten as \(y = -x+1\). Similarly, graph this line with a dashed line.
03

Shade the Solution Region

The solution to \(-1 < x+y < 1\) is the region between the two lines:- To determine which side of the line to shade, pick a test point that is not on either boundary, such as \((0,0)\).- Substitute test point into the inequality: \(-1 < 0 + 0 < 1\) holds true.- This means the region between the two dashed lines, where \(x+y\) is between -1 and 1, is the solution region. Shade this region on the graph.
04

Verify the Solution

To ensure accuracy, verify that a few points within the shaded area satisfy the inequality \(-1 < x+y < 1\). For example, points like \((0,0)\) and \((-0.5,0.5)\) should satisfy the inequality. Check these points to confirm that they lie within the correct region.

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Key Concepts

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

Absolute Value Inequality
An absolute value inequality involves absolute value expressions, which signify the distance between a number and zero on a number line. Here, the absolute value inequality is given as \(|x+y|<1\). This tells us that the sum \(x+y\) lies within 1 unit of 0 on the number line. In other words, the distance of \(x+y\) from 0 is less than 1. To solve, we convert it into a double inequality:
  • \(-1 < x+y < 1\)
This form means that \(x+y\) should be greater than \(-1\) and less than \(1\). By interpreting absolute value inequalities this way, it allows us to treat them as two simpler linear inequalities.
Coordinate Plane
To effectively graph inequalities like \(|x+y|<1\), we use the coordinate plane. The coordinate plane is a two-dimensional surface defined by a horizontal axis, known as the x-axis, and a vertical axis, known as the y-axis. Each point on this plane has a pair of coordinates, \((x, y)\), that describes its location. When dealing with inequalities, the coordinate plane helps us visualize solutions:
  • Plotting boundary lines to define regions.
  • Shading areas to show where conditions are met.
In our exercise, two boundary lines are plotted to represent the limits of the solution area. The coordinate plane thus acts as a valuable tool to see not just what the inequality represents algebraically but where the valid solutions exist spatially.
Linear Inequality
Understanding linear inequalities is crucial for graphing solutions accurately. In our case, when the inequality was rewritten as \(-1 < x+y < 1\), it consisted of two linear inequalities:
  • \(x + y > -1\)
  • \(x + y < 1\)
Each linear inequality can be represented by a line on the coordinate plane. The line for \(x+y=-1\) is expressed in slope-intercept form as \(y = -x-1\). Similarly, the line for \(x+y=1\) is \(y = -x+1\). Since we're dealing with inequalities and not just equations, the lines are dashed. This indicates points on the line aren't part of the solution, but the space between them might be.
Solution Region Shading
Solution region shading is a visual way to indicate the parts of a graph where the solutions to an inequality exist. After plotting the boundary lines for \(-1 < x+y < 1\), we need to determine which region represents the solution. We:
  • Pick a test point not on the boundary, such as \((0,0)\).
  • Check if it satisfies the inequality. In this case, it does: \(-1 < 0 < 1\).
Since the test point is valid, we know the region between the dashed boundary lines is the solution. Shading this area visually communicates where all points satisfy the inequality. Verification by plugging in additional points ensures that the shaded area accurately represents the solution set.

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Most popular questions from this chapter

(a) Digital Solutions charges for help-desk services according to the equation \(c=0.25 m+10\), where \(c\) represents the cost in dollars, and \(m\) represents the minutes of service. Complete the following table. \begin{tabular}{|l|l|l|l|l|l|l|} \hline\(m\) & 5 & 10 & 15 & 20 & 30 & 60 \\ \hline\(c\) & & & & & & \\ \hline \end{tabular} (b) Label the horizontal axis \(m\) and the vertical axis \(c\), and graph the equation \(c=0.25 m+10\) for nonnegative values of \(m\). (c) Use the graph from part (b) to approximate values for \(c\) when \(m=25,40\), and 45 . (d) Check the accuracy of your readings from the graph in part (c) by using the equation \(c=0.25 m+10\).

The equation of a line that contains the two points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) is \(\frac{y-y_{1}}{x-x_{1}}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\). We often refer to this as the two-point form of the equation of a straight line. Use the two- point form and write the equation of the line that contains each of the indicated pairs of points. Express final equations in standard form. (a) \((1,1)\) and \((5,2)\) (b) \((2,4)\) and \((-2,-1)\) (c) \((-3,5)\) and \((3,1)\) (d) \((-5,1)\) and \((2,-7)\)

What does it mean to say that two points determine a line?

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An online grocery store charges for delivery based on the equation \(C=0.30 p\), where \(C\) represents the cost in dollars, and \(p\) represents the weight of the groceries in pounds. Label the horizontal axis \(p\) and the vertical axis \(C\), and graph the equation \(C=0.30 p\) for nonnegative values of \(p\).
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