Textbook Question
Solve each quadratic equation using the quadratic formula. See Example 7. -2x² + 4x + 3 = 0
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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.6.99
Solve each inequality. Give the solution set using interval notation. See Example 10.
- 4 ≤ (x + 1)/2 ≤ 5
Verified step by step guidance1
Start by writing the compound inequality clearly: \(-4 \leq \frac{x + 1}{2} \leq 5\).
To eliminate the fraction, multiply all three parts of the inequality by 2 (which is positive, so the inequality signs remain the same): \(-4 \times 2 \leq x + 1 \leq 5 \times 2\).
Simplify the multiplication: \(-8 \leq x + 1 \leq 10\).
Next, isolate \(x\) by subtracting 1 from all parts of the inequality: \(-8 - 1 \leq x + 1 - 1 \leq 10 - 1\).
Simplify the expressions to get the solution for \(x\): \(-9 \leq x \leq 9\). Express this solution in interval notation as \([-9, 9]\).
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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 it as two separate inequalities and find the values of the variable that satisfy both simultaneously. The solution is the intersection of the solution sets for each inequality.
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Solving Linear Equations
Manipulating Inequalities with Fractions
When solving inequalities involving fractions, multiply or divide both sides by the denominator carefully, ensuring it is positive to avoid reversing the inequality sign. Simplify the expression step-by-step to isolate the variable while maintaining inequality direction.
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4:02Solving Linear Equations with Fractions
Interval Notation
Interval notation expresses the solution set of inequalities using parentheses and brackets to denote open or closed intervals. Parentheses indicate values not included, while brackets include endpoints. It provides a concise way to represent all values satisfying the inequality.
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06:01i & j Notation
Related Practice
Textbook Question
Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -3(x - 6) > 2x - 2
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Textbook Question
Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. -2(x + 3) = -6(x + 7)
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Textbook Question
Solve each quadratic equation using the square root property. See Example 6. (3x - 1)² = 12
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Textbook Question
Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -2x + 8 ≤ 16
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Textbook Question
Solve each quadratic equation using the zero-factor property. See Example 5. x² - 5x + 6 = 0