Problem 29 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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. \(x+y < 1\).

Step 2: 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.

Step 3: 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.

Step 4: 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.

Key Concepts

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.