Problem 43 - A Graphical Approach to College Algebra

Question

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} x+y & \leq 9 \\ x & \leq-y^{2} \end{aligned}$$

Step-by-Step Solution

Answer

Shade the region below the line \(y = -x + 9\) and to the left of the parabola \(x = -y^2\).

Step 1: Convert Inequalities to Equations

First, convert each inequality in the system to an equation. For the first inequality, we have the equation \[ x + y = 9 \]and for the second inequality, we have\[ x = -y^2. \]This will help us plot the boundary lines of each inequality.

Step 2: Graph the Boundary Lines

Next, draw the graphs of the equations obtained in Step 1. 1. For \(x + y = 9\), rearrange the equation to the form \(y = -x + 9\). This is a straight line with a slope of \(-1\) and a y-intercept of \(9\). 2. For \(x = -y^2\), this is a parabola opening to the left with its vertex at the origin (0,0). Plot these boundary lines on the coordinate plane.

Step 3: Determine Boundary Line Directions

Since the inequalities are \(x + y \leq 9\) and \(x \leq -y^2\), the region of interest will be below or on the line \(y = -x + 9\) and on or to the left of the parabola \(x = -y^2\). Dashed or solid lines are not needed here, as the problem specifically involves blogging the solution in these regions that include the boundaries.

Step 4: Shade the Intersection Region

The solution to the system of inequalities is the region where both conditions are satisfied at the same time. Shade below the line \(y = -x + 9\) and to the left of the parabola \(x = -y^2\). This overlapping shaded area represents the solution set to the system of inequalities. Make sure this region includes and is bounded by the respective boundary.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities is a fundamental skill in algebra that involves visualizing a range of possible solutions on a graph. To better understand how to do this, it's essential to break down the steps clearly.

• **Equation Conversion:** Begin by converting each inequality to an equation. This helps establish boundary lines, which are crucial to graphing the inequalities.
• **Graphing Boundary Lines:** For a given inequality like \(x + y \leq 9\), convert it to \(x + y = 9\) to get a clear line. Graph this line on a coordinate plane.
• **Understanding Direction:** The inequality sign tells us which side of the boundary line to shade. If the inequality is \(\leq\) or \(\geq\), the line itself is included in the solution set, often represented by a solid line.

This method visually demonstrates all solutions that satisfy the inequality, not just a single point.

System of Inequalities

A system of inequalities consists of multiple inequalities solved together to find a common solution set. Here’s how you can approach it:

• **Identify Boundaries:** Convert each inequality to an equation to determine the boundary lines, as shown in single inequality graphing.
• **Graph Each Boundary:** Draw the lines or curves for each equation on the same coordinate plane. For instance, if you have \(x + y \leq 9\) and \(x \leq -y^2\), both lines must be represented.
• **Determine the Overlapping Region:** The solution set for the system is where the shaded regions of each individual inequality overlap. This overlap represents all values that satisfy every inequality in the system.

Intersecting solutions create a clear picture of which areas satisfy both conditions, helping you find the complete solution set.

Coordinate Plane

The coordinate plane is a two-dimensional plane where each point is defined by a pair of numbers, usually referred to as coordinates. Understanding this concept is vital for graphing and solving equations or inequalities.- **Axes:** The coordinate plane is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical). - **Plotting Points:** Each point on this plane is defined as \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position. This system allows us to accurately plot lines or curves.- **Graphing Lines and Curves:** Linear equations form straight lines, while non-linear equations can form curves like parabolas. Understanding the shape helps plot each accurately.By understanding the coordinate plane, you can effectively graph inequalities or systems of equations, portraying a visual representation of potential solutions.