Textbook Question
Solve each inequality. Give the solution set using interval notation. See Example 10. -9 ≤ x + 5 ≤ 15
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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.6.75
Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. 5x +2 ≤ -48
Verified step by step guidance1
Start by isolating the variable term on one side of the inequality. Subtract 2 from both sides to get: \(5x + 2 - 2 \leq -48 - 2\).
Simplify both sides of the inequality: \(5x \leq -50\).
Next, divide both sides of the inequality by 5 to solve for \(x\). Since 5 is positive, the inequality direction remains the same: \(x \leq \frac{-50}{5}\).
Simplify the fraction to find the inequality for \(x\): \(x \leq -10\).
Express the solution set in interval notation. Since \(x\) is less than or equal to \(-10\), the solution set is \((-\infty, -10]\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solving Linear Inequalities
A linear inequality involves an inequality sign (<, ≤, >, ≥) with a linear expression. To solve it, isolate the variable by performing inverse operations, similar to solving linear equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
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Solving Linear Equations
Interval Notation
Interval notation is a way to represent solution sets of inequalities using intervals. It uses parentheses () for values not included and brackets [] for values included, indicating the range of possible solutions on the number line.
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Properties of Inequalities
Understanding how inequalities behave under addition, subtraction, multiplication, and division is crucial. Adding or subtracting the same number keeps the inequality direction, but multiplying or dividing by a negative number reverses it, which affects the solution set.
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Related Practice
Textbook Question
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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Textbook Question
Solve each quadratic equation using the quadratic formula. See Example 7. 2x² - x - 28 = 0
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Textbook Question
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 Example 10. -5 < 5 + 2x < 11
Textbook Question
Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 2(x - 8) = 3x - 16