State whether the given equation is true for all values of the variables (disregard any value that makes a denominator zero)
equation one
(2/ 4 + x) = (1/2) + (2/x)
equation two
(1 + x + x^2 / x) = (1/x) + 1 + x
one: no (try x=2)
two: yes
To determine whether the given equations are true for all values of the variables (excluding values that make a denominator zero), we can use algebraic manipulation to simplify the equations and check if the resulting expressions are equivalent.
Equation One:
(2/4 + x) = (1/2) + (2/x)
Step 1: Simplify both sides of the equation by finding the least common denominator (LCD). The LCD in this case is 4x.
(2x/x + 4x/4x) = (2/x + 2*2/4x)
(2x + x) / 4x = (2 + 4) / (4x)
Step 2: Combine like terms and simplify further.
3x / 4x = 6 / 4x
Step 3: Cross-multiply and simplify.
3x * 4x = 6 * 4x
12x^2 = 24x
Step 4: Rearrange the equation to one side.
12x^2 - 24x = 0
Step 5: Factor out the common factor.
12x(x - 2) = 0
Now, we have two cases to consider:
Case 1: x = 0
If x = 0, the equation is undefined because it would make the denominator zero. So, we disregard this value.
Case 2: x - 2 = 0
Solving for x, we get x = 2.
Since the equation is satisfied when x = 2, but not when x = 0 (as it would create a zero denominator), the given equation is not true for all values of the variable x (disregarding values that make the denominator zero).
Equation Two:
(1 + x + x^2 / x) = (1/x) + 1 + x
Step 1: Simplify both sides by finding the least common denominator (LCD). The LCD in this case is x.
(x + x^2 + x^2) / x = (1 + x + x^2) / x
Step 2: Combine like terms.
2x^2 + x = 1 + x + x^2
Step 3: Rearrange the equation to one side.
2x^2 + x - 1 - x - x^2 = 0
Step 4: Simplify and combine like terms.
x^2 + x - 1 = 0
Step 5: Factor the quadratic equation.
(x + 1)(x - 1) = 0
Now, we have two cases to consider:
Case 1: x + 1 = 0
Solving for x, we get x = -1.
Case 2: x - 1 = 0
Solving for x, we get x = 1.
Since both x = -1 and x = 1 do not make any denominators zero, we can say that the given equation is true for these values. Therefore, the given equation is true for all values of the variable x except for the values that make a denominator zero.