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Chapter 3: Problem 6

In Exercises 5 - 12, determine whether each \( x \)-value is a solution (or an approximate solution) of the equation. \( 2^{3x + 1} = 32 \) (a) \( x = -1 \) (b) \( x = 2 \)

Short Answer

Expert verified
Neither \(x = -1\) nor \(x = 2\) is a solution to the equation \(2^{3x + 1} = 32\).

Step by step solution

01

Checking Solution for \(x = -1\)

Substitute \(x = -1\) into the equation. This gives us \(2^{3*(-1) + 1} = 32\), which simplifies to \(2^{-2} = 32\). Further breaking it down we get \(1/4 = 32\). This is not true, hence \(x = -1\) is not a solution to the equation.
02

Checking Solution for \(x = 2\)

Substitute \(x = 2\) into the equation. This gives us \(2^{3*2 + 1} = 32\), which simplifies to \(2^{7} = 32\). We can rewrite 32 as \(2^5\), so if both sides of the equation are equal, then they should simplify to the same power of 2. Because 7 is not equal to 5, \(x = 2\) is not a solution to the equation.

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Key Concepts

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

Solution Verification
Verifying solutions for equations is an important skill that helpsensure our answers are correct. One way to verify solutions is by substituting the potential solution back into the original equation to see if it makes the equation true. Let's use the equation \(2^{3x + 1} = 32\) as an example.
  • Substitute the suggested \(x\)-value into the equation.
  • Simplify both sides of the equation.
  • Check whether both sides are equal.
For instance, in our exercise, we try values \(x = -1\) and \(x = 2\).If when substituting either of these into the equation, both sides become equal, that means the \(x\)-value is indeed a solution.However, if after substitution the equation does not hold, then that \(x\)-value is not a solution.
Exponential Functions
Exponential functions are functions where the variable appears in the exponent.They are written in the form \(f(x) = a^{bx+c}\), where \(a\) is a constant, and \(b\) and \(c\) are also constants.In these functions, even slight changes in the variable \(x\) cause significant changes in the value of the function.
In the equation \(2^{3x + 1} = 32\), consider the exponential part \(2^{3x + 1}\). The base here is \(2\).The power, or exponent, changes depending on the value of \(x\).
  • Exponential growth happens when the base is greater than 1.
  • Exponential decay occurs when the base is between 0 and 1.
Exponential functions powerfully model many real-life situations like population growth and radioactive decay.
Power of a Number
The power of a number tells us how many times to multiply the number by itself.For instance, the power in \(2^{3x + 1}\) represents the exponent that follows the base of 2.Manipulating powers is crucial when simplifying equations like \(2^{3x + 1} = 32\).
For instance, to see if \(2^{7} = 32\), we need to see if both sides have the same power when expressed as powers of 2.
  • \(32\) can be written as \(2^5\).
  • To have equality, we need \(2^{3x + 1}\) to simplify to \(2^5\).
The equation's balance is achieved when the powers are equal, but here \(7eq5\), therefore \(x = 2\) is not the solution.

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Most popular questions from this chapter

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The populations \( P \) (in thousands) of Horry County, South Carolina from \( 1970 \) through \( 2007 \) can be modeled by \( P = -18.5 + 92.2e^{0.0282t} \) where \( t \) represents the year, with \( t = 0 \) corresponding to 1970.(Source: U.S. Census Bureau) (a) Use the model to complete the table. (b) According to the model, when will the population of Horry County reach \( 300,000 \)? (c) Do you think the model is valid for long-term predictions of the population? Explain.
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