Problem 45 - Elementary Algebra

Question

$$ \text { For Problems 37-56, graph each linear inequality. (Objective } 3 \text { ) } $$ $$ y>-x $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Inequality

We are given the inequality \( y > -x \). This represents a region on the coordinate plane where all the points above the line \( y = -x \) satisfy the inequality.

Step 2: Graph the Boundary Line

First, we graph the line \( y = -x \). This is a straight line with a slope of -1 and a y-intercept of 0 (it passes through the origin). Since the inequality is 'greater than' and not 'greater than or equal to', this line is dashed, indicating that points on the line itself are not included in the solution.

Step 3: Determine the Shaded Region

Since the inequality is \( y > -x \), we shade the region above the line \( y = -x \). This region includes all points where the y-coordinate is greater than \(-x\).

Step 4: Verify with a Test Point

Choose a test point that is not on the line, such as (0,1). Substitute into the inequality: is \( 1 > -(0) \) ? Yes, 1 > 0, so the test point works. Thus, the shaded region is correct.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional space defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane is represented by a pair of numbers \( (x, y) \). The x-coordinate tells us the distance of the point from the y-axis, while the y-coordinate tells us the distance from the x-axis. In this space, we can graph a variety of mathematical phenomena, including linear inequalities.
To effectively understand and graph linear inequalities like \( y > -x \), it is crucial to be familiar with navigating the coordinate plane. Typically, the intersection of the axes, known as the origin, is where both the x-coordinate and y-coordinate are zero \( (0, 0) \). The coordinate plane allows us to visualize equations and inequalities as regions or lines, making it easier to identify all possible solutions that satisfy an inequality.

Boundary Line

In graphing linear inequalities, the "boundary line" is a key concept. It marks the edge of the region where the inequality holds true. For our exercise, the boundary line is represented by the equation \( y = -x \). This particular line has a slope of -1 and passes through the origin of the coordinate plane.
Since the inequality is a strict one, \( y > -x \), the boundary line is dashed rather than solid. A dashed line indicates that points lying precisely on the line do not satisfy the inequality, which is why they are excluded from the shaded region. Drawing the boundary line correctly is crucial as it divides the plane into regions – one of which will be our solution region where the inequality holds true.

Inequality Graphing

Graphing an inequality involves not just drawing its boundary line but also identifying and shading the region that satisfies the inequality. For the inequality \( y > -x \), once the boundary line is drawn as a dashed line \( y = -x \), the next step is to determine which side to shade.
Since the inequality sign \( '>' \) indicates "greater than," we shade the area above the boundary line on the coordinate plane. This shaded region includes all points that have a higher y-value than what the boundary line \( y = -x \) offers. By shading this area, we visually express all the (x, y) points that are solutions to \( y > -x \). Choosing a test point, such as \( (0, 1) \), helps confirm that this region is indeed the correct one since substitution clearly shows it satisfies the inequality.

Slope and Y-Intercept

Understanding the components of a linear equation \( y = mx + b \), such as slope and y-intercept, is essential when dealing with linear inequalities. The slope (\