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Magazine Chapter 8: Problem 51
$$ \text { For Problems 37-56, graph each linear inequality. (Objective } 3 \text { ) } $$ $$ y \leq \frac{1}{2} x-2 $$
Short Answer
Expert verified
Graph \( y = \frac{1}{2}x - 2 \) with a solid line and shade below.
Step by step solution
01
Identify the Inequality Type
The inequality given is a linear inequality of the form \( y \leq \frac{1}{2}x - 2 \). This inequality indicates the region below the line \( y = \frac{1}{2}x - 2 \).
02
Convert to an Equation for Graphing
To graph the inequality, first consider the related equation \( y = \frac{1}{2}x - 2 \). This will help you identify the boundary line of the inequality.
03
Graph the Boundary Line
Find two points to draw the line. Start with the y-intercept where \( x = 0 \), which gives \( y = -2 \). Another point can be found by selecting \( x = 2 \), leading to \( y = -1 \). Connect points (0, -2) and (2, -1) with a solid line, as the inequality is \( \leq \).
04
Determine Shaded Region
Because the inequality is \( y \leq \frac{1}{2}x - 2 \), shade the region below the line to indicate all the points \( (x, y) \) that satisfy the inequality.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Equations
A linear equation is a fundamental concept in algebra. It is an equation that forms a straight line when graphed on a coordinate plane. The standard format of a linear equation is \( y = mx + b \), where:
- \( m \) is the slope of the line, which shows how steep the line is.
- \( b \) is the y-intercept, the point where the line crosses the y-axis.
Inequality Graphing
Graphing inequalities is an extension of graphing linear equations. Instead of representing a single line, inequalities such as \( y \leq \frac{1}{2}x - 2 \) illustrate a shaded region on the graph that includes all the solutions. To graph an inequality:
- Start by graphing the boundary line as if it were an equation.
- Choose a solid line if the inequality includes equal to (\( \leq \) or \( \geq \)). Use a dashed line if it does not (\( < \) or \( > \)).
- Shade the region where the inequality holds true. For \( \leq \) or \( < \), shade below the line. For \( \geq \) or \( > \), shade above.
Solving Inequalities
Solving inequalities involves finding the range of values for which the inequality holds true. Unlike equations, inequalities do not provide one fixed solution but a set of possible solutions. Here鈥檚 how you solve the inequality \( y \leq \frac{1}{2}x - 2 \):
- Treat the inequality as an equation to find the boundary line initially.
- Identify if the inequality includes (\( \leq \)) or excludes (\( < \)) the boundary.
- Use test points to determine which side of the boundary satisfies the inequality if needed.
Coordinate Plane
The coordinate plane is a two-dimensional surface where each point is defined by a pair of numerical coordinates \((x, y)\). The horizontal line is the x-axis, and the vertical line is the y-axis. Understanding and using this plane is a core element in graphing.
- The point where the two axes meet is the origin, denoted as \((0, 0)\).
- Coordinates are used to plot points, graph lines, and define regions.
- The plane enables visual representation, which aids in comprehending relationships among equations and inequalities.
Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It provides a way to represent real-world problems through equations and inequalities. Skills developed in algebra are used throughout mathematics and science. Key components include:
- Understanding variables as symbols that stand for unknown values.
- Using algebraic methods to solve equations and inequalities.
- Recognizing patterns and relations to formulate mathematical expressions.
