Textbook Question
Solve each inequality. Give the solution set using interval notation. See Example 10.
- 4 ≤ (x + 1)/2 ≤ 5
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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.6.81
Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -3(x - 6) > 2x - 2
Verified step by step guidance1
Start by distributing the -3 on the left side of the inequality: write the expression as \(-3 \times (x - 6)\) and apply the distributive property to get \(-3x + 18\).
Rewrite the inequality with the distributed terms: \(-3x + 18 > 2x - 2\).
Next, collect all the variable terms on one side and the constant terms on the other side. For example, add \$3x$ to both sides and add \(2\) to both sides to isolate terms: \(18 + 2 > 2x + 3x\).
Simplify both sides: combine like terms to get \(20 > 5x\).
Finally, solve for \(x\) by dividing both sides of the inequality by 5, remembering to keep the inequality direction the same since you are dividing by a positive number. Express the solution set in interval notation based on the inequality obtained.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solving Linear Inequalities
Linear inequalities involve expressions with variables raised to the first power and inequality signs. To solve them, isolate the variable on one side by performing algebraic operations, similar to solving equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
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Solving Linear Equations
Distributive Property
The distributive property allows you to multiply a single term across terms inside parentheses, i.e., a(b + c) = ab + ac. Applying this property correctly is essential to simplify expressions before solving inequalities.
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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, clearly showing the range of possible solutions on the number line.
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Related Practice
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Solve each quadratic equation using the quadratic formula. See Example 7. -2x² + 4x + 3 = 0
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Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -2x + 8 ≤ 16
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Solve each quadratic equation using the zero-factor property. See Example 5. x² - 100 = 0
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Solve each quadratic equation using the zero-factor property. See Example 5. x² - 5x + 6 = 0