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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 64
Chapter 6, Problem 64

Graph the solution set of each system of inequalities.
y≥3xy\(\ge\)3^{x}
y≥2y\(\ge\)2

Verified step by step guidance
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Step 1: Identify the inequalities in the system. The system consists of two inequalities: \( y \geq 3^x \) and \( y \geq 2 \).
Step 2: Graph the first inequality \( y \geq 3^x \). Start by graphing the boundary line \( y = 3^x \), which is an exponential function. Plot points for several values of \( x \) to get the shape of the curve, such as \( (0, 1), (1, 3), (2, 9) \), and so on. Since the inequality is \( \geq \), shade the region above the curve.
Step 3: Graph the second inequality \( y \geq 2 \). This is a horizontal line at \( y = 2 \). Since the inequality is \( \geq \), shade the region above this line.
Step 4: Determine the solution set by finding the intersection of the shaded regions from both inequalities. The solution set is where the shaded region from \( y \geq 3^x \) overlaps with the shaded region from \( y \geq 2 \).
Step 5: Clearly mark the solution set on the graph. This is the region that satisfies both inequalities, and it will be the area above both the curve \( y = 3^x \) and the line \( y = 2 \).

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Key Concepts

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

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols such as '≥' (greater than or equal to) and '≤' (less than or equal to) to indicate the range of possible solutions. Understanding how to interpret and graph inequalities is crucial for visualizing solution sets in coordinate systems.
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Graphing Exponential Functions

Exponential functions, such as y = 3^x, represent growth or decay processes and are characterized by a constant base raised to a variable exponent. When graphing these functions, it is important to recognize their shape, which typically includes a horizontal asymptote and increases rapidly for positive x-values. This understanding helps in determining the region of the graph that satisfies the inequality.
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Graphs of Exponential Functions

Shading Regions in Graphs

When graphing inequalities, shading is used to represent the solution set. For a given inequality, the area above (or below) the line is shaded to indicate all the points that satisfy the inequality. Knowing how to correctly shade the regions based on the inequality symbols is essential for accurately depicting the solution set on a graph.
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Guided course
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Graphs & the Rectangular Coordinate System
Related Practice
Textbook Question

Answer each question. A line passes through the points of intersection of the graphs of y = x2 and x2 + y2 = 90. What is the equation of this line?

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.

4x + 3y = -7

2x + 3y = -11

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Textbook Question

Graph the solution set of each system of inequalities.

y≤logxy\(\le\]\log\) x

y≥∣x−2∣y\(\ge\]\left\)|x-2\(\right\)|

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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

Solve each problem. Find the equation of the line passing through the points of intersection of the graphs of x2 + y2 = 20 and x2 - y = 0.

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