Question

guided practice
solve \\(\frac{1}{x} + \frac{x}{3} = \frac{4}{3}\\).
the lcd is 3x. now use it to solve the equation.
type your answer and then click or tap done.
solve the equation.
x=\boxed{} or x=\boxed{}

Explanation:

Step1: Subtract $\frac{x}{3}$ from both sides

$\frac{1}{x} = \frac{4}{3} - \frac{x}{3}$

Step2: Combine right-hand terms

$\frac{1}{x} = \frac{4 - x}{3}$

Step3: Cross-multiply to eliminate fractions

$3 = x(4 - x)$

Step4: Rearrange into quadratic form

$x^2 - 4x + 3 = 0$

Step5: Factor the quadratic

$(x - 1)(x - 3) = 0$

Step6: Solve for x

$x = 1$ or $x = 3$

Wait, correcting the initial step (original equation: $\frac{1}{x} + \frac{x}{3} = \frac{4}{3}$):

Step1: Multiply all terms by LCD $3x$

$3x \cdot \frac{1}{x} + 3x \cdot \frac{x}{3} = 3x \cdot \frac{4}{3}$

Step2: Simplify each term

$3 + x^2 = 4x$

Step3: Rearrange into standard quadratic

$x^2 - 4x + 3 = 0$

Step4: Factor

$(x - 1)(x - 3) = 0$

Step5: Find roots

$x = 1$ or $x = 3$

Final Answer: $x = 1$ or $x = 3$

Step1: Multiply by LCD $3x$

$3 + x^2 = 4x$

Step2: Rearrange to quadratic form

$x^2 - 4x + 3 = 0$

Step3: Factor the quadratic

$(x - 1)(x - 3) = 0$

Step4: Solve for x

$x = 1$ or $x = 3$

Answer:

$\frac{1}{2}$