Textbook Question
Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -2x + 8 ≤ 16
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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.6.95
Solve each inequality. Give the solution set using interval notation. See Example 10. 10 ≤ 2x + 4 ≤ 16
Verified step by step guidance1
Start by understanding that the compound inequality \(10 \leq 2x + 4 \leq 16\) means that \(2x + 4\) is simultaneously greater than or equal to 10 and less than or equal to 16.
To isolate \(x\), subtract 4 from all parts of the inequality: \(10 - 4 \leq 2x + 4 - 4 \leq 16 - 4\), which simplifies to \(6 \leq 2x \leq 12\).
Next, divide all parts of the inequality by 2 to solve for \(x\): \(\frac{6}{2} \leq \frac{2x}{2} \leq \frac{12}{2}\), which simplifies to \(3 \leq x \leq 6\).
Interpret the solution: \(x\) is greater than or equal to 3 and less than or equal to 6.
Express the solution set in interval notation as \([3, 6]\), which includes both endpoints.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Compound Inequalities
A compound inequality involves two inequalities joined together, such as 'a ≤ expression ≤ b'. Solving it requires isolating the variable so that the inequality holds true for both parts simultaneously, resulting in a range of values.
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Finding the Domain and Range of a Graph
Solving Linear Inequalities
Solving linear inequalities involves performing algebraic operations like addition, subtraction, multiplication, or division to isolate the variable. When multiplying or dividing by a negative number, the inequality sign must be reversed.
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Interval Notation
Interval notation is a concise way to represent solution sets of inequalities using parentheses and brackets. Brackets [ ] indicate inclusion of endpoints, while parentheses ( ) indicate exclusion, clearly showing the range of valid values.
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