Problem 48 - Intermediate Algebra

Question

$$ \text { Graph the system: }\left\\{\begin{array}{l} {x \geq 0} \\ {y \geq 0} \\ {y \geq x^{2}+1} \\ {y \leq 4-x} \end{array}\right. $$

Step-by-Step Solution

Answer

Graph each inequality and shade the overlapping first quadrant region.

Step 1: Identify the Constraints

First, examine each inequality in the system. The constraints are given by: \( x \geq 0 \), \( y \geq 0 \), \( y \geq x^2 + 1 \), and \( y \leq 4 - x \). We need to graph these inequalities and find the region that satisfies all of them simultaneously.

Step 2: Graph the X and Y Axes Constraints

Plot the lines \( x = 0 \) and \( y = 0 \) on the graph. These represent the constraints \( x \geq 0 \) and \( y \geq 0 \) respectively. The solution region must lie in the first quadrant of the coordinate plane, where both \( x \) and \( y \) are positive.

Step 3: Graph the Quadratic Inequality

Next, plot the quadratic inequality \( y \geq x^2 + 1 \). Begin by graphing the parabola \( y = x^2 + 1 \). This parabola opens upwards and has its vertex at the point (0, 1). The region above this parabola represents the inequality \( y \geq x^2 + 1 \). Shade the region above and on the parabola.

Step 4: Graph the Linear Inequality

Plot the linear inequality \( y \leq 4 - x \). Start by drawing the line \( y = 4 - x \). This line goes through points (0, 4) and (4, 0) with a negative slope of -1. The area below this line signifies the inequality \( y \leq 4 - x \). Shade the region below and on this line.

Step 5: Determine the Feasible Region

The feasible region is the area where the shaded regions from Steps 3 and 4 overlap in the first quadrant (where \( x \geq 0 \) and \( y \geq 0 \)). This is the region that satisfies all given inequalities. Look for the intersection of the shaded areas above the parabola \( y = x^2 + 1 \) and below the line \( y = 4 - x \) in the first quadrant.

Key Concepts

Graphing Inequalities

Graphing inequalities involves visually representing regions in the coordinate plane that satisfy an inequality's condition.
This is typically done by shading areas in the graph. Let's break it down:

• Start by identifying the boundary lines of the inequality, either given by a linear equation or a curve like a parabola.
• Plot these lines on the graph. If the inequality is strict (e.g., greater than or less than), use a dashed line. If it includes equality (e.g., greater than or equal to), use a solid line.
• Determine which side of the line or curve should be shaded. This shows all the points that satisfy the inequality. A simple test point, often the origin, can help determine the correct side to shade.

Combining these steps for each inequality in a system helps to find the overlapping region where all inequalities hold true simultaneously.

Quadratic Inequalities

Quadratic inequalities involve expressions where variables are squared, resulting in parabolic graphs. These inequalities are slightly more complex to graph than linear ones.

• Begin by graphing the corresponding quadratic equation, such as the equation of the parabola defining the boundary.
• The inequality sign indicates whether you shade above or below the parabola. For instance, if the inequality is of the form \( y \geq x^2 + c \), shade above the parabola.
• The vertex of the parabola is a crucial point, as it is often where more complex boundaries and intersections can occur.

After plotting and shading, the region representing the solution to the quadratic inequality can be seen on the graph.

Linear Inequalities

Linear inequalities are simpler, involving straight lines.
To graph them:

• Start with the boundary line from the linear equation. If given \( y \leq mx + b \), plot the line \( y = mx + b \) using its slope \(m\) and y-intercept \(b\).
• Decide which side to shade. For instance, shade below for \( y \leq mx + b \).
• If the inequality involves "less than or equal to", use a solid line; otherwise, use a dashed line.

The solution region for the linear inequality will be a half-plane on one side of the line, and when graphed with other inequalities, it helps define the feasible region.

Feasible Region

The feasible region is the area on the graph where all solutions to the system of inequalities exist. Finding this region involves identifying where all the conditions of a system overlap.

• Graph each inequality as described above.
• The feasible region is where all the shaded areas of the individual inequalities intersect.
• This region must satisfy the conditions of all inequalities simultaneously.

In problems like our exercise, this often results in a bounded area, such as a triangle or another polygon shape. In real-world scenarios, the feasible region can represent possible solutions or configurations, like optimal configurations in linear programming.