Problem 4 - Algebra and Trigonometry

Question

\(3-16=\) Graph the inequality. $$ y \geq-2 $$

Step-by-Step Solution

Answer

Graph the line \(y = -2\) and shade above it.

Step 1: Solve the Equation

Calculate the difference of the numbers given: \(3 - 16 = -13\). This is not directly related to the inequality but checks your understanding of solving arithmetic problems.

Step 2: Understand the Inequality

The inequality \(y \geq -2\) means that the value of \(y\) is greater than or equal to \(-2\). We need to represent this on a graph.

Step 3: Graph the Boundary Line

Draw a horizontal line on the graph at \(y = -2\). Since the inequality is "greater than or equal to," this line will be solid to include points where \(y = -2\).

Step 4: Shade the Solution Area

In the graph, shade the area above the line \(y = -2\). This represents all the points where \(y > -2\), including those on the line itself where \(y = -2\).

Key Concepts

Inequalities

Inequalities are a crucial mathematical concept. They describe the relationships between values that aren't precisely equal. In essence, inequalities let you know if one number is larger, smaller, or possibly equal to another.
They are denoted by symbols like:

• \(<\) for "less than"
• \(>\) for "greater than"
• \(\leq\) for "less than or equal to"
• \(\geq\) for "greater than or equal to"

These symbols provide insight into a range of solutions rather than just a single one.
For instance, the inequality \(y \geq -2\) suggests that \(y\) can be any number greater than \(-2\), including \(-2\) itself. This is different from an equation, where you would solve for exact numbers and outcomes. Inequalities represent more flexible solutions and possibilities.
Understanding inequalities is key when you want to describe situations where there can be multiple permissible outcomes.

Graphing Linear Equations

Graphing linear equations is an essential skill in mathematics, allowing you to visualize relationships between variables. Linear equations are simple and usually take the form \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept.
To graph such an equation, you begin by plotting the y-intercept

• identify the point on the y-axis where \(y = b\)

Next, use the slope \(m\) to determine the rise over run. This tells you how far up or down to go and how far left or right at each step from your initial point.
Drawing a straight line through these points helps you capture all potential solutions to the equation.
Graphing provides a visual depiction of numerical relationships. It translates abstract numbers into something you can see and understand more clearly.
Graphing these equations helps not just in solving them, but also in understanding how changes in variables affect outcomes.

Linear Inequalities

Linear inequalities combine the properties of linear equations with inequalities. They express a linear relation with a range of possible solutions.
The inequality \(y \geq -2\) is one example:

• The boundary line here is simple: \(y = -2\). This line is part of the solution, hence it's drawn solid.
• Shading symbolizes where solutions exist; here, it's above the line, representing \(y > -2\) and including \(y = -2\).

To graph a linear inequality, begin with the boundary line, just like graphing a linear equation.
When graphing:

• A solid line is used for "\(\leq\)" or "\(\geq\)" inequalities, indicating the line itself is part of the solution.
• A dashed line is used to indicate a strict "\(<\)" or "\(>\)" inequality, meaning the points on the line aren't solutions.

By graphing inequalities, you create a visual representation of all the potential solutions in a way that's easy to understand and analyze.