x/x – 1 - 3/x = 1/2
I am not sure how to solve this equation and show my work. My problem just says to solve the equation.
Thanks.
Begin by putting parentheses in the right places. It is ambiguous as written. That means that it can be interpreted in different ways. x/x = 1, for example
To solve the equation (x/x – 1) - (3/x) = 1/2, we can follow these steps:
Step 1: Simplify the expression
First, let's simplify the expression by finding a common denominator for the fractions. The common denominator for x/x and 3/x is x, so we can rewrite the equation as:
(x/x – 1)(x/x) - (3/x)(x/x) = 1/2
This simplifies to:
(x^2 - x) / x - (3x) / x = 1/2
Simplifying further:
(x^2 - x - 3x) / x = 1/2
Step 2: Combine like terms
Next, let's combine like terms in the numerator:
(x^2 - 4x) / x = 1/2
Step 3: Clear fractions
To clear the fraction on the left side of the equation, we can multiply both sides of the equation by 2x:
2x * (x^2 - 4x) / x = 2x * 1/2
This simplifies to:
2(x^2 - 4x) = x
Step 4: Distribute and simplify
Distribute 2 to both terms in the parentheses:
2x^2 - 8x = x
Step 5: Move all terms to one side
To solve for x, let's move all terms to one side of the equation by subtracting x from both sides:
2x^2 - 8x - x = 0
This simplifies to:
2x^2 - 9x = 0
Step 6: Factor the equation
To factor the quadratic equation, let's find two numbers that multiply to give -18 (2 * -9) and add to give -9. Since -3 and 6 satisfy these conditions, we can write the equation as:
(2x + 3)(x - 6) = 0
Step 7: Solve for x
To find the possible values of x, we can set each factor equal to zero and solve for x:
2x + 3 = 0 or x - 6 = 0
For the first equation, when we solve for x, we get:
2x = -3
x = -3/2
For the second equation, we get:
x = 6
So the equation x/x – 1 - 3/x = 1/2 has two solutions: x = -3/2 and x = 6.