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Chapter 8: Problem 8

In Exercises \(5-20,\) graph each linear inequality. $$y > -x$$

Short Answer

Expert verified
Graph \( y = -x \) as a dashed line and shade above it.

Step by step solution

01

Understand the Inequality

The inequality given is \( y > -x \). This inequality represents a region in the coordinate plane where \( y \) is greater than the value of \(-x\). In this context, we need to distinguish which side of the boundary line \( y = -x \) is shaded.
02

Graph the Boundary Line

First, plot the boundary line \( y = -x \). This line is a straight line with a slope of \(-1\) and a y-intercept of \(0\). To do this, choose points such as \((0,0)\) and \((1,-1)\) to draw the line. Since the inequality is \( y > -x \), this line will be dashed to indicate that the points on the line are not included in the solution.
03

Test a Point

We need to determine which side of the boundary line to shade. A simple way to do this is to test a point not on the line, such as \((0,1)\). Substitute into the inequality: \( 1 > -(0) \), which simplifies to \( 1 > 0 \). This statement is true, meaning our test point is in the solution region.
04

Shade the Solution Region

Since the test point \((0,1)\) satisfies the inequality, shade the region above the boundary line \( y = -x \). This region contains all points \((x,y)\) such that \( y > -x \).

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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 way to visually represent all solutions of an inequality on a coordinate plane. An inequality might look something like this: \( y > -x \). This means we are looking for all points \((x, y)\) where \( y \) is greater than \( -x \). When graphing linear inequalities, the process consists of plotting the boundary line and then identifying which side of this line the solution region falls on.

Here's how to proceed with graphing linear inequalities:
  • First, convert the inequality into an equation by replacing the inequality sign with an equals sign. For example, \( y = -x \).
  • Next, graph this equation to serve as the boundary line.
  • Determine if the boundary line should be solid or dashed. A dashed line is used if the inequality symbol is \( > \) or \( < \), indicating points on the line are not included in the solution. Use a solid line if the inequality is \( \geq \) or \( \leq \).
  • Finally, pick a test point not on the boundary line to decide which side of the line to shade. This shading represents all solutions to the inequality.
By visually showing where the solutions lie, one can easily understand and interpret the inequality.
Coordinate Plane
The coordinate plane is a two-dimensional surface on which we graph points, lines, and regions. It consists of two perpendicular lines called axes. The horizontal axis is the x-axis, and the vertical axis is the y-axis.

Important features of the coordinate plane include:
  • The point where the x-axis and y-axis intersect is called the origin, denoted as \((0,0)\).
  • The plane is divided into four sections called quadrants.
  • Points are represented by pairs \((x, y)\), where \(x\) is the horizontal distance from the origin, and \(y\) is the vertical distance.
Understanding how to navigate the coordinate plane is crucial when plotting equations or inequalities. As in our example with \( y > -x \), we use the coordinate plane to graph this inequality to understand which regions satisfy the inequality.
Boundary Line
The boundary line is critical when representing a linear inequality, functioning as the dividing line on the coordinate plane. For the inequality \( y > -x \), the equation of the boundary line is \( y = -x \). It essentially tells us where to start placing the solution.

This line is drawn based on the following steps:
  • Calculate two or more points that satisfy the boundary line equation. For \( y = -x \), points like \((0,0)\) and \((1,-1)\) are chosen.
  • Plot these points on the coordinate plane and draw a straight line through them.
  • Decide the line style. In "\( y > -x \)," the use of a dashed line indicates excluding the line's points from the inequality solution.
In this manner, the boundary line serves as the reference for separating the solution region from the non-solution region on the graph.
Solution Region
After establishing the boundary line, determining the solution region involves shading the side of the line where the inequality holds true. In the case of \( y > -x \), we need to locate and shade the region where \( y \) values are greater than \( -x \).

To find this solution region:
  • Choose a test point that is not on the boundary line, such as \((0,1)\). Substitute the coordinates into the inequality to check if they satisfy it.
  • For \( y > -x \), plugging in gives \( 1 > -0 \) which is true. Thus, the side where this point lies represents the solution region.
  • Shade this side of the boundary line to visually indicate all points satisfying the inequality.
The solution region includes an infinite number of points. It is visually represented on the graph, which aids in understanding the inequality's implications by showing where all possible solutions exist.

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