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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 43
Chapter 6, Problem 43

Graph the solution set of each system of inequalities.
4x - 3y ≤ 12
y ≤ x2

Verified step by step guidance
1
Step 1: Identify each inequality in the system. The first inequality is \(4y - 2x \leq 15\) and the second inequality is \(y \geq -x^{2} + 2\).
Step 2: Rewrite the first inequality in slope-intercept form (\(y = mx + b\)) to make graphing easier. Start by isolating \(y\): \(4y \leq 2x + 15\), then divide both sides by 4 to get \(y \leq \frac{1}{2}x + \frac{15}{4}\).
Step 3: Graph the line \(y = \frac{1}{2}x + \frac{15}{4}\). Since the inequality is \(\leq\), shade the region below or on this line.
Step 4: Graph the parabola \(y = -x^{2} + 2\). Since the inequality is \(y \geq -x^{2} + 2\), shade the region above or on this parabola.
Step 5: The solution set to the system is the region where the shaded areas from both inequalities overlap. This overlapping region satisfies both inequalities simultaneously.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Linear Inequalities

A linear inequality like 4y - 2x ≤ 15 represents a half-plane on the coordinate plane. To graph it, first rewrite it in slope-intercept form (y ≤ (1/2)x + 15/4), then draw the boundary line (solid for ≤ or ≥) and shade the region that satisfies the inequality.
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Graphing Quadratic Inequalities

A quadratic inequality such as y ≥ -x² + 2 involves a parabola. The graph of y = -x² + 2 is a downward-opening parabola shifted up by 2 units. The inequality y ≥ -x² + 2 means shading the region on or above this parabola.
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Solution Set of a System of Inequalities

The solution set of a system of inequalities is the intersection of the regions satisfying each inequality. Graph each inequality separately, then identify the overlapping shaded area that meets all conditions simultaneously.
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Related Practice
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Let A=[−2403]A = \(\left\)[ \(\begin{matrix}\) -2 & 4 \\ 0 & 3 \(\end{matrix}\) \(\right\)] and B=[−6240]B = \(\left\)[ \(\begin{matrix}\) -6 & 2 \\ 4 & 0 \(\end{matrix}\) \(\right\)]. Find each of the following.

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Let A=[−2403]A = \(\left\)[ \(\begin{matrix}\) -2 & 4 \\ 0 & 3 \(\end{matrix}\) \(\right\)] and B=[−6240]B = \(\left\)[ \(\begin{matrix}\) -6 & 2 \\ 4 & 0 \(\end{matrix}\) \(\right\)]. Find each of the following.

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