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Chapter 6: Problem 27

Solve. If no solution exists, state this. $$ \frac{2}{6}+\frac{1}{2 x}=\frac{1}{3} $$

Short Answer

Expert verified
No solution exists.

Step by step solution

01

- Simplify the fractions

Simplify the fraction \(\frac{2}{6} \) to \(\frac{1}{3} \). Thus, the equation becomes: \[ \frac{1}{3} + \frac{1}{2x} = \frac{1}{3} \]
02

- Subtract fractions on both sides

To isolate \(\frac{1}{2x} \), subtract \(\frac{1}{3} \) from both sides of the equation: \[ \frac{1}{3} + \frac{1}{2x} - \frac{1}{3} = \frac{1}{3} - \frac{1}{3} \] This simplifies to: \[ \frac{1}{2x} = 0 \]
03

- Solve for x

Since \(\frac{1}{2x} = 0 \), the only solution is if the denominator is infinite, which is not possible for a finite value of \(x\). Therefore, the equation has no solution.

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Key Concepts

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

simplifying fractions
To simplify fractions, you need to find the greatest common divisor (GCD) of the numerator and the denominator. Then, divide both by the GCD.
In our example, we began with the fraction \(\frac{2}{6}\). The GCD of 2 and 6 is 2.
When we divide both the numerator and the denominator by 2, we get:
\( \frac{2 梅 2}{6 梅 2} = \frac{1}{3} \).
Thus, simplifying fractions means to reduce them to their simplest form.
isolating variables
Isolating variables means rearranging an equation so that the variable you are solving for is by itself on one side of the equation.
In our case, after simplifying the fractions, the equation became: \(\frac{1}{3} + \frac{1}{2x} = \frac{1}{3} \).
To isolate \(\frac{1}{2x}\), we need to remove \(\frac{1}{3}\) from both sides of the equation.
We achieve this by subtracting \(\frac{1}{3} \) from both sides:
\( \frac{1}{3} + \frac{1}{2x} - \frac{1}{3} = \frac{1}{3} - \frac{1}{3} \).
This results in: \(\frac{1}{2x} = 0\).
Therefore, isolating the variable helps in determining its value.
non-solvable equations
Some equations have no solutions, which we call non-solvable equations.
In our example, after isolating the variable, we ended up with: \(\frac{1}{2x} = 0\).
A fraction equals zero only if its numerator is zero. However, \(1 eq 0\), indicating no real number solution for x.
Mathematically, to set the denominator \(2x\) to infinity for \(\frac{1}{2x} \) to be zero is not possible.
Therefore, we conclude that the original equation has no solution.

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