Question

decide whether the equation is an identity, a conditional equation, or a contradiction. give the solution set
24x + 6 = 3(4x - 26)

a. conditional; {-7}
b. contradiction, ∅
c. conditional; {6}
d. identity; {all real numbers}

Explanation:

Step1: Expand the right side

First, we expand the right - hand side of the equation \(24x + 6=3(4x - 26)\). Using the distributive property \(a(b + c)=ab+ac\) (here \(a = 3\), \(b = 4x\), \(c=- 26\)), we get \(3\times4x-3\times26=12x - 78\). So the equation becomes \(24x+6 = 12x-78\).

Step2: Subtract \(12x\) from both sides

Subtract \(12x\) from both sides of the equation \(24x + 6=12x-78\). We have \((24x-12x)+6=(12x - 12x)-78\), which simplifies to \(12x+6=-78\).

Step3: Subtract 6 from both sides

Subtract 6 from both sides of the equation \(12x + 6=-78\). We get \(12x+6 - 6=-78 - 6\), which simplifies to \(12x=-84\).

Step4: Divide both sides by 12

Divide both sides of the equation \(12x=-84\) by 12. \(\frac{12x}{12}=\frac{-84}{12}\), so \(x=-7\).

Since the equation is true only when \(x = - 7\) (not for all real numbers and not a contradiction), it is a conditional equation with the solution set \(\{-7\}\).

Answer:

A. Conditional; \(\{-7\}\)