Problem 5 - Precalculus with Calculus Previews

Question

Graph the given inequality. \(-y \geq 2(x+3)-5\)

Step-by-Step Solution

Answer

Graph the line \(y = -2x - 1\) and shade below it.

Step 1: Simplify the Inequality

First, let's simplify the given inequality. You have \(-y \geq 2(x+3)-5\). Distribute the 2 on the right side: \(-y \geq 2x + 6 - 5\). Simplify the right side: \(-y \geq 2x + 1\).

Step 2: Solve for y

To make it easier to graph the inequality, solve for \(y\). Multiply the entire inequality by -1, remembering to flip the inequality sign: \(y \leq -2x -1\). Now the inequality is in a more standard form for graphing.

Step 3: Graph the Boundary Line

Graph the line \(y = -2x - 1\). This line is the boundary of the solution set. Use a solid line because the inequality \(\leq\) includes the line itself. You can find points by choosing values for \(x\), such as \(x = 0\), which gives \(y = -1\), and \(x = 1\), which gives \(y = -3\). Plot these points and draw the line.

Step 4: Shade the Solution Region

Since the inequality is \(y \leq -2x - 1\), you shade the region below the line. This shaded region represents all the points \((x, y)\) that satisfy the inequality.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that create straight lines when plotted on a graph. They typically take the form \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept, the point where the line crosses the y-axis.
Understanding linear equations is foundational because they describe the relationship between two variables in a simple and straightforward manner.

• Slope \(m\): Indicates the steepness of the line and direction. A positive slope means the line is ascending, while a negative slope means it's descending.
• Y-intercept \(b\): This is the value of \(y\) when \(x\) is zero. It tells you where the line starts on the graph when moving from the y-axis.

Linear equations are used in a variety of fields, such as physics, economics, and engineering, to model relationships and predict outcomes.

Inequality Symbols

Inequality symbols are used to compare two expressions and show that one is greater or lesser in relation to the other. The basic inequality symbols include:

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

When dealing with inequalities like \(-y \geq 2x + 1\), it is important to remember to flip the inequality sign when multiplying or dividing both sides by a negative number.
This idea is crucial in solving and graphing inequalities, as it impacts the direction of the inequality and consequently the section of the graph to be shaded.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graphically represent equations and inequalities using coordinates \(x, y\). The horizontal axis is the x-axis and the vertical axis is the y-axis.
This grid allows for precise plotting of points, lines, and regions.

• Origin (0,0): The point where the x-axis and y-axis intersect, essentially the center of the coordinate plane.
• Quadrants: The plane is divided into four quadrants, each representing a different combination of positive and negative values for x and y.

Understanding the coordinate plane is vital for graphing as it provides the framework for locating points and visualizing the solution sets of inequalities.

Shaded Region

In inequality graphing, the shaded region indicates all the possible solutions that satisfy the inequality. After plotting the boundary line, which in this case is given by \(y = -2x - 1\), you must decide which side of the line contains the solutions.
If the inequality symbol is "\(\leq\)" or "\(\geq\)", use a solid line to indicate that points on the line are included in the solution.

• For "\(\leq\)" or "<", shade below the line.
• For "\(\geq\)" or ">", shade above the line.

In this exercise, the inequality \(y \leq -2x - 1\) means shading below the boundary line. This shaded area graphically represents all the point combinations of \(x\) and \(y\) that meet the inequality, revealing a visual map of solutions.