/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 Solve the inequality. $$\log (... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 63

Solve the inequality. $$\log (x-2)+\log (9-x)<1$$

Short Answer

Expert verified
The solution is \(x \in (4, 7)\).

Step by step solution

01

Combine the Logarithms

The expression \( \log (x-2) + \log (9-x) \) can be combined using the logarithmic product rule, which states that \( \log a + \log b = \log (a \times b) \). Thus, we have: \[ \log ((x-2)(9-x)) < 1 \]
02

Convert Logarithmic Inequality to Exponential Form

Recall that if \( \log_b a < c \), then \( a < b^c \). Since the logarithm is base 10 here, we write: \[ (x-2)(9-x) < 10^1 \] Simplifying, we get:\[ (x-2)(9-x) < 10 \]
03

Simplify the Quadratic Expression

Expand and simplify the expression \((x-2)(9-x)\): \[ x(9-x) - 2(9-x) = 9x - x^2 - 18 + 2x = -x^2 + 11x - 18 \]Thus, we rewrite the inequality as: \[ -x^2 + 11x - 18 < 10 \]
04

Rearrange the Inequality

Move all terms to one side to form a standard quadratic inequality:\[ -x^2 + 11x - 18 - 10 < 0 \]Simplify to: \[ -x^2 + 11x - 28 < 0 \]
05

Solve the Quadratic Equation

Solve for roots of the quadratic equation \(-x^2 + 11x - 28 = 0\) by applying the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a = -1, b = 11, c = -28\):\[ x = \frac{-11 \pm \sqrt{11^2 - 4(-1)(-28)}}{2(-1)} \]\[ x = \frac{-11 \pm \sqrt{121 - 112}}{-2} \]\[ x = \frac{-11 \pm 3}{-2} \]Roots are: \[ x_1 = 4, \quad x_2 = 7 \]
06

Test Intervals

The roots \(x = 4\) and \(x = 7\) divide the number line into three intervals: \((-\infty, 4)\), \((4, 7)\), and \((7, \infty)\). Test a value from each interval in the inequality \(-x^2 + 11x - 28 < 0\). - For \(x = 3\), \(-3^2 + 11 \cdot 3 - 28 = -9 + 33 - 28 = -4 < 0\).- For \(x = 5\), \(-5^2 + 11 \cdot 5 - 28 = -25 + 55 - 28 = 2 ot< 0\).- For \(x = 8\), \(-8^2 + 11 \cdot 8 - 28 = -64 + 88 - 28 = -4 < 0\).Thus, the solution is \((4, 7)\).
07

Consider the Domain

The domain of the original inequality limits \(x\) such that both \(x - 2 > 0\) and \(9 - x > 0\). Thus: - \(x - 2 > 0\) implies \(x > 2\). - \(9 - x > 0\) implies \(x < 9\).The domain is \(x \in (2, 9)\).
08

Determine the Overlapping Solution

Combine the interval where the inequality holds \((4, 7)\) with the domain \((2, 9)\) from Step 7. The overlap of these intervals is \((4, 7)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Product Rule
The logarithmic product rule is a crucial concept when dealing with logarithmic expressions, especially inequalities. This rule states that the sum of two logarithms with the same base can be combined into a single logarithm by multiplying their arguments. Mathematically, it's written as:
  • \( \log_b a + \log_b c = \log_b (a \times c) \)
This simplifies equations and is particularly useful in problems where you need to combine logs for simplification, just as we did with \( \log(x-2) + \log(9-x) \).
By applying the rule, we combine them to become \( \log((x-2)(9-x)) \). This simplification allows for straightforward transformation and manipulation.
Understanding this rule will not only simplify many logarithmic expressions, but it's also foundational for moving to the next step in solving logarithmic inequalities.
Quadratic Inequality
Quadratic inequalities involve expressions of the form \( ax^2 + bx + c < 0 \) or \( ax^2 + bx + c > 0 \). These expressions result from expanding a product such as \((x-2)(9-x)\) into its equivalent quadratic form. In our original exercise, we end up with:
  • \( -x^2 + 11x - 28 < 0 \)
The solution to a quadratic inequality like this involves finding the roots of the equation \( -x^2 + 11x - 28 = 0 \). After finding roots, which in this case are \( x_1 = 4 \) and \( x_2 = 7 \), we test the intervals they create on the number line.
We determine on which intervals the inequality holds true by choosing test points within these intervals.
This approach will help you understand the behavioral change of the quadratic expression across its critical points, assisting in finding the actual solution set for the inequality.
Domain of a Logarithmic Function
The domain of a logarithmic function is the set of all possible input values (\(x\)) for which the function is defined. Logarithmic functions like \( \log(x-2) \) and \( \log(9-x) \) have a domain restriction. The argument of a logarithmic function must always be a positive number, i.e., greater than zero.
  • For \( \log(x-2) \), the domain requirement is \( x-2 > 0 \), which simplifies to \( x > 2 \).
  • For \( \log(9-x) \), the domain requirement is \(9-x > 0 \), meaning \( x < 9 \).
Together, these conditionals are combined to establish the domain of the original logarithmic inequality. Thus, the valid range for \( x \) in this problem is the overlap of both conditions, resulting in \( (2, 9) \).
Understanding the domain of these functions is essential, as it determines the initial set of values that you can work with.
Without considering domain restrictions, some solutions could potentially involve undefined values, leading to incorrect conclusions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the Change of Base Formula to show that $$\log e=\frac{1}{\ln 10}$$

Radioactive iodine is used by doctors as a tracer in diagnosing certain thyroid gland disorders. This type of iodine decays in such a way that the mass remaining after \(t\) days is given by the function $$ m(t)=6 e^{-0.087 t} $$ where \(m(t)\) is measured in grams. (a) Find the mass at time \(t=0\) (b) How much of the mass remains after 20 days?

Use the Laws of Logarithms to combine the $$4 \log x-\frac{1}{3} \log \left(x^{2}+1\right)+2 \log (x-1)$$

A 50 -gallon barrel is filled completely with pure water. Salt water with a concentration of 0.3 Ib/gal is then pumped into the barrel, and the resulting mixture overflows at the same rate. The amount of salt in the barrel at time \(t\) is given by $$ Q(t)=15\left(1-e^{-0.04 t}\right) $$ where \(t\) is measured in minutes and \(Q(t)\) is measured in pounds. (a) How much salt is in the barrel after 5 min? (b) How much salt is in the barrel after 10 min? (c) Draw a graph of the function \(Q(t)\) (d) Use the graph in part (c) to determine the value that the amount of salt in the barrel approaches as \(t\) becomes large. Is this what you would expect? (IMAGES CANNOT COPY) $$Q(t)=15\left(1-e^{-0.04 t}\right)$$

When a certain medical drug is administered to a patient, the number of milligrams remaining in the patient's bloodstream after \(t\) hours is modeled by $$ D(t)=50 e^{-0.2 t} $$ How many milligrams of the drug remain in the patient's bloodstream after 3 hours?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.