Problem 44 - A Graphical Approach to College Algebra

Question

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &x+2 y \leq 4\\\ &y \geq x^{2}-1 \end{aligned}$$

Step-by-Step Solution

Answer

Graph both, find the overlap and verify with a point like (0,0).

Step 1: Understand the Inequalities

We have two inequalities: 1) Linear inequality: \( x + 2y \leq 4 \) and 2) Quadratic inequality: \( y \geq x^2 - 1 \). The goal is to determine the solution set that satisfies both inequalities.

Step 2: Graph the Linear Inequality

First, rewrite the linear equation in slope-intercept form: \( y \leq -\frac{1}{2}x + 2 \). Graph the line \( y = -\frac{1}{2}x + 2 \). Since the inequality is \( \leq \), shade the region below the line. Use a solid line because the inequality includes equal to (\( \leq \)).

Step 3: Graph the Quadratic Inequality

Graph \( y = x^2 - 1 \) as a parabola that opens upwards with vertex (0, -1). Since the inequality is \( \geq \), shade the region above the parabola. Use a solid curve because the inequality includes equal to (\( \geq \)).

Step 4: Identify the Solution Region

The solution region is where the shaded areas from both inequalities overlap. This common region represents all the points that satisfy both \( x + 2y \leq 4 \) and \( y \geq x^2 - 1 \).

Step 5: Test a Point

As a final check, test a point within the overlapping region, such as \((0, 0)\). Substitute into both inequalities to confirm it satisfies them. \(0 + 2(0) = 0 \leq 4\) (True), and \(0 \geq 0^2 - 1\) gives \(0 \geq -1\) (True), confirming it lies in the solution region.

Key Concepts

Graphing Inequalities

Graphing inequalities involves the visual representation of regions that satisfy a given inequality condition. The purpose is to show all possible solutions within a coordinate plane.
Here’s how you can approach it:

• Identify the type of curve or line you’re dealing with (linear or quadratic).
• Write the equation in a form that is easy to graph. For linear inequalities, the slope-intercept form is common: \[y = mx + b\]. For quadratic inequalities, the standard form is \[y = ax^2 + bx + c\].

Once the equation is in the correct form, graph the line or curve.
Use a solid line if the inequality includes equal to (\( \leq \) or \( \geq \)) and a dashed line if it does not.
After drawing the line, shade the appropriate region on the graph:

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

Finally, the solution set is the region where the shaded areas of all inequalities intersect. This region shows all the solutions that satisfy every inequality in the system.

Linear Inequality

A linear inequality looks similar to a linear equation but uses inequality symbols (\(<, \leq, >\), and \(\geq\)) instead of an equals sign.
Linear inequalities can describe a wide range of scenarios.
To visualize a linear inequality:

• First, transform the inequality into slope-intercept form \(y = mx + b\), if necessary. This makes it easy to graph.
• Next, draw the line by plotting the y-intercept (\(b\)) and using the slope (\(m\)) to find another point.
• Determine whether to draw a solid or dashed line. Use a solid line if the inequality involves \( \leq \) or \( \geq \), and a dashed line for \( < \) or \( > \).

After drawing the line, choose a test point not on the line (often \((0,0)\) if not on the line) to decide which side of the line to shade.
If the point satisfies the inequality, shade the region including that point; otherwise, shade the opposite side.
The shaded area represents all solutions of the inequality.

Quadratic Inequality

Quadratic inequalities involve quadratic expressions, typically seen as \(y > ax^2 + bx + c\) or \(y < ax^2 + bx + c\).
Understanding the graph helps visualize the set of possible solutions.
Here’s how you can graph it step-by-step:

• Identify the vertex of the parabola, where the graph has a peak or trough. For \(y = ax^2 + bx + c\), the vertex form can be useful, or use the formula: vertex \((h, k)\) where \(h = -\frac{b}{2a}\) and \(k\) is found by substituting \(h\) back into the equation.
• Plot the vertex and several other points obtained by substituting different values for \(x\).
• Draw the curve. It can be a solid line if the inequality is \( \geq \) or \( \leq \), or a dashed curve for strict inequalities \( > \) or \( < \).

Shade the region that satisfies the inequality:

• Above the parabola for \(\geq\) or \(>\)
• Below the parabola for \(\leq\) or \(<\)

The overlapped shaded region when combined with others will give the set of solutions that satisfy the entire system of inequalities. This process ensures all potential solutions within the dimensional restrictions are considered.