Textbook Question
Solve each quadratic equation using the quadratic formula. See Example 7.
x² - x - 1 = 0
Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem R.6.91
Solve each inequality. Give the solution set using interval notation. See Example 10. -9 ≤ x + 5 ≤ 15
Verified step by step guidance1
Start with the compound inequality: \(-9 \leq x + 5 \leq 15\).
To isolate \(x\), subtract 5 from all three parts of the inequality: \(-9 - 5 \leq x + 5 - 5 \leq 15 - 5\).
Simplify each part: \(-14 \leq x \leq 10\).
Interpret the solution: \(x\) is greater than or equal to \(-14\) and less than or equal to \(10\).
Express the solution set in interval notation as \([-14, 10]\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Compound Inequalities
Compound inequalities involve two inequalities combined into one statement, such as -9 ≤ x + 5 ≤ 15. Solving them requires isolating the variable by performing operations on all parts of the inequality simultaneously to maintain the inequality's truth.
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Properties of Inequalities
When solving inequalities, adding, subtracting, multiplying, or dividing all parts by the same positive number preserves the inequality direction. Multiplying or dividing by a negative number reverses the inequality signs. Understanding these properties ensures correct manipulation of inequalities.
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Interval Notation
Interval notation expresses solution sets of inequalities concisely using brackets and parentheses. Brackets [ ] indicate inclusion of endpoints, while parentheses ( ) indicate exclusion. For example, [a, b] means all values from a to b, including a and b.
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Related Practice
Textbook Question
Solve each quadratic equation using the quadratic formula. See Example 7. x² - 6x = -7
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Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 4(2x + 7) = 2x + 22 + 3(2x + 2)
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Solve each quadratic equation using the zero-factor property. See Example 5. 5x² - 3x - 2 = 0
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Solve each quadratic equation using the quadratic formula. See Example 7. x² - 4x + 3 = 0
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Textbook Question
Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. 5x +2 ≤ -48
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