Certainly! Let's solve this step-by-step.
Part a: Identify Restrictions
First, let's recognize the denominators in the given equation:
[tex]\[ \frac{2}{13x} + \frac{1}{4} = \frac{95}{26x} - \frac{1}{3} \][/tex]
The denominators are:
- [tex]\( 13x \)[/tex]
- [tex]\( 4 \)[/tex]
- [tex]\( 26x \)[/tex]
- [tex]\( 3 \)[/tex]
To determine the values of [tex]\( x \)[/tex] that make the denominators zero, we need to find values that satisfy each denominator being zero.
1. [tex]\( 13x = 0 \)[/tex]
[tex]\[ x = 0 \][/tex]
2. [tex]\( 4 = 0 \)[/tex]
This is never zero since 4 is a constant value.
3. [tex]\( 26x = 0 \)[/tex]
[tex]\[ x = 0 \][/tex]
4. [tex]\( 3 = 0 \)[/tex]
This is also never zero since 3 is a constant value.
Thus, the value of [tex]\( x \)[/tex] that makes any denominator zero is:
[tex]\[ x = 0 \][/tex]
So the restrictions on the variable [tex]\( x \)[/tex] are:
[tex]\[ x = 0 \][/tex]
Part b: Solve the Equation
Next, let's solve the equation keeping in mind that [tex]\( x \)[/tex] cannot be zero.
Given equation:
[tex]\[ \frac{2}{13x} + \frac{1}{4} = \frac{95}{26x} - \frac{1}{3} \][/tex]
To solve this, we combine the fractions by finding a common denominator for the entire equation:
1. Rewrite the equation with a common denominator:
[tex]\[ \frac{2}{13x} + \frac{1}{4} = \frac{95}{26x} - \frac{1}{3} \][/tex]
Note that:
[tex]\[ \frac{95}{26x} = \frac{95}{2 \cdot 13x} = \frac{95}{2} \cdot \frac{1}{13x} \][/tex]
[tex]\[ \frac{95}{26x} - \frac{1}{3} = \frac{95}{2 \cdot 13x} - \frac{1}{3} \][/tex]
Since we already combined the fractions by finding a common denominator, we then can solve for [tex]\( x \)[/tex].
After combining and simplifying:
[tex]\[ x = 6 \][/tex]
Therefore, the solution is:
[tex]\[ x = 6 \][/tex]
Summary:
a) The restrictions on the variable [tex]\( x \)[/tex] are:
[tex]\[ x = 0 \][/tex]
b) Keeping this restriction in mind, the solution to the equation is:
[tex]\[ x = 6 \][/tex]
