Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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![**Equation Solution**
To solve the equation for \( x \):
\[ 2 \ln(x-3) = \ln(x+5) + \ln 4 \]
**Steps to Solve:**
1. **Apply the Power Rule**: Use the power rule for logarithms, which states that \( a \ln b = \ln b^a \). Therefore, rewrite the left side:
\[ \ln((x-3)^2) = \ln(x+5) + \ln 4 \]
2. **Use the Addition Property**: For the right side, use the property that \( \ln a + \ln b = \ln(ab) \).
\[ \ln((x-3)^2) = \ln((x+5) \cdot 4) \]
\[ \ln((x-3)^2) = \ln(4(x+5)) \]
3. **Equate the Arguments**: Since the logarithms are equal, their arguments must also be equal.
\[ (x-3)^2 = 4(x+5) \]
4. **Expand and Simplify**:
\[ x^2 - 6x + 9 = 4x + 20 \]
5. **Rearrange the Equation**:
\[ x^2 - 6x + 9 - 4x - 20 = 0 \]
\[ x^2 - 10x - 11 = 0 \]
6. **Solve the Quadratic Equation**: Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = -10 \), and \( c = -11 \).
\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot (-11)}}{2 \cdot 1} \]
\[ x = \frac{10 \pm \sqrt{100 + 44}}{2} \]
\[ x = \frac{10 \pm \sqrt{144}}{2} \]
\[ x = \frac{10 \pm 12}{2} \]
7. **Calculate the Two Possible Values**:
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a799525-bc02-4aa7-a724-19d6c1834605%2Faac6c6b8-f99f-40b7-a119-524daefef732%2F358rpgj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Equation Solution**
To solve the equation for \( x \):
\[ 2 \ln(x-3) = \ln(x+5) + \ln 4 \]
**Steps to Solve:**
1. **Apply the Power Rule**: Use the power rule for logarithms, which states that \( a \ln b = \ln b^a \). Therefore, rewrite the left side:
\[ \ln((x-3)^2) = \ln(x+5) + \ln 4 \]
2. **Use the Addition Property**: For the right side, use the property that \( \ln a + \ln b = \ln(ab) \).
\[ \ln((x-3)^2) = \ln((x+5) \cdot 4) \]
\[ \ln((x-3)^2) = \ln(4(x+5)) \]
3. **Equate the Arguments**: Since the logarithms are equal, their arguments must also be equal.
\[ (x-3)^2 = 4(x+5) \]
4. **Expand and Simplify**:
\[ x^2 - 6x + 9 = 4x + 20 \]
5. **Rearrange the Equation**:
\[ x^2 - 6x + 9 - 4x - 20 = 0 \]
\[ x^2 - 10x - 11 = 0 \]
6. **Solve the Quadratic Equation**: Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = -10 \), and \( c = -11 \).
\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot (-11)}}{2 \cdot 1} \]
\[ x = \frac{10 \pm \sqrt{100 + 44}}{2} \]
\[ x = \frac{10 \pm \sqrt{144}}{2} \]
\[ x = \frac{10 \pm 12}{2} \]
7. **Calculate the Two Possible Values**:
\[
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