Question

solve the equation, and check the solution.
\\(\frac{1}{3}(5x + 3) - \frac{1}{4}(x + 5) = 4\\)

select the correct choice below and, if necessary, fill in the answer box to complete your answer
a. the solution set is \\(\\{\\}\\)

b. the solution set is \\{all real numbers\\}

c. the solution set is \\(\varnothing\\)

Explanation:

Step1: Eliminate denominators

Multiply both sides by 12 (LCM of 3 and 4) to get:
$$12\times\frac{1}{3}(5x + 3)-12\times\frac{1}{4}(x + 5)=12\times4$$
Simplify each term:
$$4(5x + 3)-3(x + 5)=48$$

Step2: Distribute the coefficients

Expand the brackets:
$$20x + 12-3x - 15 = 48$$

Step3: Combine like terms

Combine the x - terms and constant terms:
$$(20x-3x)+(12 - 15)=48$$
$$17x-3 = 48$$

Step4: Solve for x

Add 3 to both sides:
$$17x=48 + 3$$
$$17x=51$$
Divide both sides by 17:
$$x=\frac{51}{17}$$
$$x = 3$$

Step5: Check the solution

Substitute \(x = 3\) into the original equation:
Left - hand side:
$$\frac{1}{3}(5\times3+3)-\frac{1}{4}(3 + 5)$$
$$=\frac{1}{3}(15 + 3)-\frac{1}{4}(8)$$
$$=\frac{1}{3}\times18-2$$
$$=6 - 2=4$$
Right - hand side is 4. Since LHS = RHS, \(x = 3\) is a valid solution.

Answer:

A. The solution set is \(\{3\}\)