Question

solve the equation, if possible. check the result. (if there is no solution, enter no solution. if all real numbers are\\
\\(\frac{3(b + 5)}{2} = \frac{3b - 12}{4}\\)\\
b = -14\\
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Explanation:

Step1: Cross - multiply to eliminate fractions

Given the equation \(\frac{3(b + 5)}{2}=\frac{3b-12}{4}\), cross - multiplying (multiplying both sides by the least common multiple of 2 and 4, which is 4) gives \(4\times\frac{3(b + 5)}{2}=4\times\frac{3b - 12}{4}\).
Simplifying the left - hand side: \(2\times3(b + 5)=6(b + 5)\), and the right - hand side: \(3b-12\). So the equation becomes \(6(b + 5)=3b-12\).

Step2: Expand the left - hand side

Using the distributive property \(a(b + c)=ab+ac\), where \(a = 6\), \(b=b\) and \(c = 5\), we get \(6b+30=3b-12\).

Step3: Subtract \(3b\) from both sides

Subtracting \(3b\) from both sides of the equation \(6b + 30=3b-12\) gives \(6b-3b+30=3b-3b - 12\), which simplifies to \(3b+30=-12\).

Step4: Subtract 30 from both sides

Subtracting 30 from both sides: \(3b+30 - 30=-12-30\), so \(3b=-42\).

Step5: Divide both sides by 3

Dividing both sides of the equation \(3b=-42\) by 3 gives \(b=\frac{-42}{3}=-14\).

Step6: Check the solution

Substitute \(b = - 14\) into the original equation:
Left - hand side: \(\frac{3(-14 + 5)}{2}=\frac{3\times(-9)}{2}=\frac{-27}{2}\)
Right - hand side: \(\frac{3\times(-14)-12}{4}=\frac{-42-12}{4}=\frac{-54}{4}=\frac{-27}{2}\)
Since the left - hand side equals the right - hand side when \(b=-14\), the solution is valid.

Answer:

\(b=-14\)