Chapter 3: Problem 234
For \(x \in R, 3^{72}(1 / 3)^{x}(1 / 3)^{\sqrt{x}}>1\) then (a) \(x \in[0,64]\) (b) \(x \in(0,64)\) (c) \(x \in[0,64)\) (d) None of these
Short Answer
Expert verified
(d) None of these
Step by step solution
01
Simplify the inequality
First, let's simplify the given inequality: \(3^{72}(1 / 3)^{x}(1 / 3)^{\sqrt{x}}>1\)
Since \(1/3 = 3^{-1}\), we can rewrite the inequality as:
\(3^{72}3^{-x}3^{-\sqrt{x}}>1\)
Now, we can combine the exponential terms:
\(3^{(72-x-\sqrt{x})}>1\)
02
Eliminate the exponent
Now, let's eliminate the exponent by taking the natural logarithm (ln) of both sides:
\(\ln{(3^{(72-x-\sqrt{x})})}>\ln{1}\)
The logarithmic property states that \(\ln{a^b} = b\ln{a}\). Therefore, we have:
\((72-x-\sqrt{x})\ln{3}>0\)
Since \(\ln{3}\) is a positive number, the inequality doesn't change:
\(72-x-\sqrt{x}>0\)
03
Solve for x
Now, move the x terms to the right side:
\(\sqrt{x}>72-x\)
Now let's square both sides to get rid of the square root:
\((\sqrt{x})^2 > (72-x)^2\)
\(x > 72^2 - 144x + x^2\)
Next, move all the terms to the left side:
\(0 > x^2 - 145x + 72^2\)
Now, let's find the range of x for which the inequality holds:
\(0 > (x - 64)(x - 81)\)
Here, we see that the inequality holds when x is between 64 and 81. Now, let's compare this range to the given options.
04
Compare the range to the given options
Looking at the options:
(a) x ∈ [0,64]
(b) x ∈ (0,64)
(c) x ∈ [0,64)
(d) None of these
None of the options provided match the range that we found (x ∈ (64,81)), so the correct answer is:
(d) None of these
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Inequalities
Exponential inequalities are a fascinating aspect of algebra that involve expressions with variable exponents. These expressions take the form of base raised to a power, where the power itself contains the variable. For example, an inequality like this may look like \(3^x > 2\) or more complex ones like \(3^{72-x-\sqrt{x}}>1\). The goal is to find the range of values of the variable that satisfy the inequality.
Exponential inequalities often require us to simplify expressions to make them more manageable. A common strategy is to express terms with the same base, which allows us to apply properties of exponents more easily. In our example, terms like \((1/3)^x\) can be rewritten as \(3^{-x}\), facilitating further manipulations.
Once simplified, we can often apply logarithmic functions to eliminate exponents, solving for the variable in question. This involves using the property \(\ln{a^b} = b\ln{a}\), which transforms the problem into a more straightforward linear inequality.
To solve these, one must carefully apply algebraic properties and sometimes even more advanced techniques like squaring both sides, as required in the given problem. Attention to detail in steps ensures accurate solutions.
Exponential inequalities often require us to simplify expressions to make them more manageable. A common strategy is to express terms with the same base, which allows us to apply properties of exponents more easily. In our example, terms like \((1/3)^x\) can be rewritten as \(3^{-x}\), facilitating further manipulations.
Once simplified, we can often apply logarithmic functions to eliminate exponents, solving for the variable in question. This involves using the property \(\ln{a^b} = b\ln{a}\), which transforms the problem into a more straightforward linear inequality.
To solve these, one must carefully apply algebraic properties and sometimes even more advanced techniques like squaring both sides, as required in the given problem. Attention to detail in steps ensures accurate solutions.
Solving Inequalities
Solving inequalities involves finding the range of possible values that satisfy the given conditions. Unlike equations, inequalities illustrate multiple solutions rather than a single point. They use symbols like \(>\), \(<\), \(\geq\), and \(\leq\) to show the relation between two expressions.
Solving inequalities often involves several algebraic manipulations. Firstly, simplifying the inequality by combining like terms or factoring can help. In our original exercise's solution, the inequality was simplified from \(3^{72}(1/3)^x(1/3)^{\sqrt{x}} > 1\) to \(72-x-\sqrt{x}>0\).
Once simplified, solving inequalities can require graphing the expressions or testing intervals for validity. In handling quadratic inequalities, as seen in the problem, one might need to rearrange terms and even apply the quadratic formula to find critical points.
Often, you'll need to consider the direction of the inequality sign, as flipping might occur when multiplying or dividing by negative numbers. Paying careful attention to these details is crucial since any mistakes can lead to incorrect solutions.
Solving inequalities often involves several algebraic manipulations. Firstly, simplifying the inequality by combining like terms or factoring can help. In our original exercise's solution, the inequality was simplified from \(3^{72}(1/3)^x(1/3)^{\sqrt{x}} > 1\) to \(72-x-\sqrt{x}>0\).
Once simplified, solving inequalities can require graphing the expressions or testing intervals for validity. In handling quadratic inequalities, as seen in the problem, one might need to rearrange terms and even apply the quadratic formula to find critical points.
Often, you'll need to consider the direction of the inequality sign, as flipping might occur when multiplying or dividing by negative numbers. Paying careful attention to these details is crucial since any mistakes can lead to incorrect solutions.
Domain of Real Functions
The domain of real functions refers to the set of input values for which the function is defined. Understanding the domain is crucial when dealing with equations and inequalities involving real numbers. It establishes the permissible range of values for variables that ensure the function remains valid.
For example, the inequality \(3^{72-x-\sqrt{x}}>1\) contains a square root. The domain of this expression requires \(x\) to be non-negative since the square root function is only defined for non-negative real numbers, making the lower bound of \(x\geq0\).
In solving inequalities, identifying these domains early on prevents mathematical errors. It ensures that all variable transformations respect the allowable input ranges, avoiding undefined expressions.
Typically, these assessments involve examining potential restrictions arising from roots, denominators, or logarithmic expressions. In solving the step-by-step exercise, recognizing the domain could aid in understanding why specific solutions fall outside the expected range of options, offering insights into which choices are viable and ensuring logical consistency.
For example, the inequality \(3^{72-x-\sqrt{x}}>1\) contains a square root. The domain of this expression requires \(x\) to be non-negative since the square root function is only defined for non-negative real numbers, making the lower bound of \(x\geq0\).
In solving inequalities, identifying these domains early on prevents mathematical errors. It ensures that all variable transformations respect the allowable input ranges, avoiding undefined expressions.
Typically, these assessments involve examining potential restrictions arising from roots, denominators, or logarithmic expressions. In solving the step-by-step exercise, recognizing the domain could aid in understanding why specific solutions fall outside the expected range of options, offering insights into which choices are viable and ensuring logical consistency.
