Textbook Question
In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. 2x = 3y + 4 4x = 3 - 5y
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Ch. 5 - Systems of Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 6, Problem 41
Graph the solution set of each system of inequalities or indicate that the system has no solution.
Verified step by step guidance1
Step 1: Identify the system of inequalities:
\[\begin{cases} -2x + y > 6 \\ -2x + y < -4 \end{cases}\]
Step 2: Rewrite each inequality in slope-intercept form (solve for y):
For the first inequality:
\[-2x + y > 6 \implies y > 2x + 6\]
For the second inequality:
\[-2x + y < -4 \implies y < 2x - 4\]
Step 3: Graph the boundary lines for each inequality:
- For \[y = 2x + 6\], draw a dashed line because the inequality is strict ( > ).
- For \[y = 2x - 4\], also draw a dashed line for the same reason ( < ).
Step 4: Determine the solution region for each inequality:
- For \[y > 2x + 6\], shade the region above the line.
- For \[y < 2x - 4\], shade the region below the line.
Step 5: Find the intersection of the two shaded regions. Since the first region is above the line \[y = 2x + 6\] and the second is below \[y = 2x - 4\], check if there is any overlap. Because \[2x + 6\] is always greater than \[2x - 4\], there is no region where both inequalities are true simultaneously, so the system has no solution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Graphing Linear Inequalities
Graphing linear inequalities involves plotting the boundary line represented by the corresponding linear equation and then shading the region that satisfies the inequality. The boundary line is dashed if the inequality is strict (>, <) and solid if it includes equality (≥, ≤). This visual representation helps identify all points that satisfy the inequality.
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System of Inequalities and Solution Sets
A system of inequalities consists of two or more inequalities considered simultaneously. The solution set is the intersection of the regions satisfying each inequality. If no common region exists, the system has no solution. Understanding how to find and interpret this intersection is key to solving such problems.
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Converting Inequalities to Slope-Intercept Form
To graph inequalities easily, rewrite them in slope-intercept form (y = mx + b). This form clearly shows the slope and y-intercept, making it easier to draw the boundary line. For example, from -2x + y > 6, isolate y to get y > 2x + 6, which guides the graphing process.
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Related Practice
Textbook Question
In Exercises 29–42, solve each system by the method of your choice.
Textbook Question
In Exercises 43–46, let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The sum of two numbers is 10 and their product is 24. Find the numbers.
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Textbook Question
Perform each long division and write the partial fraction decomposition of the remainder term. (x5+2)/(x2-1)
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Textbook Question
Graph the solution set of each system of inequalities or indicate that the system has no solution.
Textbook Question
Write the partial fraction decomposition of each rational expression. (4x2+3x+14)/(x3 - 8)
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