Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice. 1/x + 1/(x + 3) = 1/4
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Ch. 1 - 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 2, Problem 106
Use the table to solve each inequality. - 3 < 2x - 5 ≤ 3
Verified step by step guidance1
Start with the compound inequality: \(3 < 2x - 5 \leq 3\).
Break the compound inequality into two separate inequalities:
1) \(3 < 2x - 5\)
2) \(2x - 5 \leq 3\).
Solve the first inequality \(3 < 2x - 5\) by adding 5 to both sides: \(3 + 5 < 2x\), which simplifies to \(8 < 2x\). Then divide both sides by 2 to isolate \(x\): \(4 < x\).
Solve the second inequality \(2x - 5 \leq 3\) by adding 5 to both sides: \(2x \leq 8\). Then divide both sides by 2: \(x \leq 4\).
Combine the two results to find the solution to the compound inequality: \(4 < x \leq 4\). Since this is a contradiction (no number is both greater than 4 and less than or equal to 4), check the table values for \(x\) and \(y = 2x - 13\) to verify if any \(x\) satisfies the original inequality.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solving Compound Inequalities
A compound inequality involves two inequalities joined by 'and' or 'or'. To solve, treat each inequality separately and find the intersection (for 'and') or union (for 'or') of their solution sets. For example, solving 3 < 2x - 5 ≤ 3 requires isolating x in both inequalities and combining the results.
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Linear Inequalities
Using Tables to Solve Inequalities
Tables can help solve inequalities by providing values of expressions for specific x-values. By comparing the expression values to the inequality bounds, you can identify which x-values satisfy the inequality. This method is useful when direct algebraic manipulation is complex or when verifying solutions.
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Linear Functions and Their Graphs
A linear function like y = 2x - 13 produces a straight line when graphed. Understanding how changes in x affect y helps solve inequalities involving linear expressions. The table shows corresponding y-values for given x-values, illustrating the function's behavior and aiding in solving inequalities.
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Related Practice
Textbook Question
Solve each equation. - 2{7 - [4 -2(1 - x) + 3]} = 10 - [4x - 2(x - 3)]
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Textbook Question
When 3 times a number is subtracted from 4, the absolute value of the difference is at least 5. Use interval notation to express the set of all numbers that satisfy this condition.
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Textbook Question
In Exercises 104–106, express each interval in set-builder notation and graph the interval on a number line. (-2, ∞)
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Textbook Question
Solve each equation. 4x + 13 - {2x - [4(x - 3) - 5]} = 2(x - 6)
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Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice. 1/x + 1/(x + 2) = 1/3
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