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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 62
Chapter 6, Problem 62

Graph the solution set of each system of inequalities.
y≤x3−xy\(\le\) x^3-x
y>−3y>-3

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Identify the two inequalities in the system: \(y \leq x^3 - x\) and \(y > -3\).
Graph the boundary curve for the first inequality, which is the cubic function \(y = x^3 - x\). Since the inequality is \(\leq\), include the curve itself as a solid line.
Shade the region below or on the curve \(y = x^3 - x\) because the inequality is \(y \leq x^3 - x\).
Graph the horizontal boundary line \(y = -3\). Since the inequality is strict (\(y > -3\)), draw this line as a dashed line to indicate points on the line are not included.
Shade the region above the line \(y = -3\) because the inequality is \(y > -3\). The solution set is the overlap of the shaded regions from both inequalities.

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Key Concepts

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

Graphing Inequalities

Graphing inequalities involves shading the region of the coordinate plane that satisfies the inequality. For example, y ≤ x³ - x means shading all points on or below the curve y = x³ - x. The boundary curve is included if the inequality is ≤ or ≥, and excluded if it is < or >.
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Cubic Functions and Their Graphs

A cubic function like y = x³ - x has an S-shaped curve with turning points. Understanding its shape helps in accurately sketching the boundary for the inequality. Key features include intercepts, local maxima and minima, and end behavior as x approaches ±∞.
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Graphs of Logarithmic Functions

Systems of Inequalities

A system of inequalities requires finding the region where all inequalities overlap. For y ≤ x³ - x and y > -3, the solution set is the area below the cubic curve but above the line y = -3. The final graph shows the intersection of these shaded regions.
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Related Practice
Textbook Question

Solve each problem. Find all values of b such that the straight line 3x - y = b touches the circle x2 + y2 = 25 at only one point.

Textbook Question

Find each product, if possible.

[312532][3−6430]\(\left\)[ \(\begin{matrix}\) \(\sqrt{3}\) & 1 \\ 2\(\sqrt{5}\) & 3\(\sqrt{2}\) \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) \(\sqrt{3}\) & -\(\sqrt{6}\) \\ 4\(\sqrt{3}\) & 0 \(\end{matrix}\) \(\right\)]

Textbook Question

Solve each problem using a system of equations in two variables. See Example 6. The longest side of a right triangle is 13 m in length. One of the other sides is 7 m longer than the shortest side. Find the lengths of the two shorter sides of the triangle.

Textbook Question

Answer each question. Does the straight line 3x - 2y = 9 intersect the circle x2 + y2 = 25? (Hint: To find out, solve the system formed by these two equations.)

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Textbook Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

x + y = 4

2x - y = 2

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Textbook Question
Find each product, if possible. See Examples 5–7. <4x2 Matrix>
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