Problem 51 - Elementary Algebra

Question

$$ \text { For Problems 37-56, graph each linear inequality. (Objective } 3 \text { ) } $$ $$ y \leq \frac{1}{2} x-2 $$

Step-by-Step Solution

Answer

Graph \( y = \frac{1}{2}x - 2 \) with a solid line and shade below.

Step 1: 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 \( y = \frac{1}{2}x - 2 \).

Step 2: 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.

Step 3: 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 \( \leq \).

Step 4: 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.

Key Concepts

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.

Understanding linear equations is crucial because they represent constant rates of change, making them applicable in many real-life situations like calculating speed or creating budgets. Recognizing the slope and y-intercept helps you graph the line and predict outcomes.

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.

This visual representation aids in understanding which points on the graph fulfill the inequality's condition.

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’s 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.

These steps help solidify the solution set, providing insights into what values satisfy the inequality's condition.

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.

Familiarity with the coordinate plane is crucial for efficiently graphing 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.

Mastery in algebra sets the groundwork for advancing into more complex calculus and higher-level math, where these basic principles are applied to solve intricate problems.