Textbook Question
Solve each quadratic equation using the zero-factor property. See Example 5.
x² + 2x - 8 = 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.6.35
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
Verified step by step guidance1
Start by expanding the left side of the equation: multiply 2 by each term inside the parentheses to get \(2(x - 8) = 2x - 16\).
Rewrite the equation with the expanded left side: \(2x - 16 = 3x - 16\).
Next, isolate the variable terms on one side by subtracting \$2x\( from both sides: \(2x - 16 - 2x = 3x - 16 - 2x\), which simplifies to \)-16 = x - 16$.
Then, isolate \(x\) by adding 16 to both sides: \(-16 + 16 = x - 16 + 16\), which simplifies to \(0 = x\).
Interpret the result: since \(x = 0\) is the only solution, the equation is a conditional equation with the solution set \(\{0\}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Types of Equations: Identity, Conditional, and Contradiction
An identity is an equation true for all values of the variable, a conditional equation is true only for specific values, and a contradiction has no solution. Recognizing these types helps determine the nature of the solution set.
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Solving Linear Equations
Solving linear equations involves isolating the variable by applying inverse operations such as addition, subtraction, multiplication, or division. This process helps find the values that satisfy the equation.
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Checking Solutions and Solution Sets
After solving, substituting the solution back into the original equation verifies its validity. The solution set includes all values that satisfy the equation, which can be a single value, all real numbers, or an empty 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 Examples 8 and 9. 5x +2 ≤ -48
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Textbook Question
Solve each inequality. Give the solution set using interval notation. See Example 10. -5 < 5 + 2x < 11