Textbook Question
Solve each quadratic equation using the quadratic formula. See Example 7. x² - 6x = -7
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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.6.33
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)
Verified step by step guidance1
Start by expanding both sides of the equation to simplify the expressions. Use the distributive property: multiply 4 by each term inside the parentheses on the left side, and multiply 3 by each term inside the parentheses on the right side. This gives you: \(4(2x + 7) = 2x + 22 + 3(2x + 2)\) becomes \(8x + 28 = 2x + 22 + 6x + 6\).
Next, combine like terms on the right side of the equation. Add the terms involving \(x\) and the constant terms separately: \(2x + 6x = 8x\) and \(22 + 6 = 28\). So the equation now looks like \(8x + 28 = 8x + 28\).
Now, analyze the simplified equation. Since both sides are identical expressions, this suggests the equation might be an identity. To confirm, subtract \(8x + 28\) from both sides to see if the equation reduces to a true statement.
After subtracting, you get \(0 = 0\), which is always true regardless of the value of \(x\). This means the original equation holds for all real numbers \(x\).
Therefore, conclude that the equation is an identity, and the solution set is all real numbers, often written as \(\{ x \mid x \in \mathbb{R} \}\).
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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 simplifying both sides, combining like terms, and isolating the variable. This process helps find the values that satisfy the equation or determine if no or infinite solutions exist.
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Solution Sets
The solution set is the collection of all values that satisfy the equation. For identities, it includes all real numbers; for conditional equations, specific values; and for contradictions, it is empty.
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Related Practice
Textbook Question
Solve each inequality. Give the solution set using interval notation. See Example 10. -9 ≤ x + 5 ≤ 15
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Textbook Question
Solve each quadratic equation using the zero-factor property. See Example 5. 5x² - 3x - 2 = 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
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