Question

Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 4(x + 7) = 2(x + 12) + 2(x + 1)

Accepted Answer

Start by expanding both sides of the equation to simplify the expressions. Use the distributive property: multiply 4 by each term inside the parentheses on the left side, and similarly multiply 2 by each term inside the parentheses on the right side. After expanding, combine like terms on each side of the equation to simplify further. This means adding or subtracting terms with the variable $$ x $$ and constant terms separately. Next, get all terms involving $$ x  on $$one side of the equation and all constant terms on the other side. You can do this by adding or subtracting terms from both sides to isolate $$ x . $$Once $$ x  is $$isolated, simplify the equation to see if you get a true statement for all values of $$ x $$, a specific value of $$ x $$, or no solution. This will help determine if the equation is an identity, conditional equation, or contradiction. Finally, based on the simplified form, state the solution set: if the equation is true for all $$ x $$, the solution set is all real numbers (identity); if true for specific $$ x $$, list those values (conditional); if no values satisfy it, the solution set is empty (contradiction).

Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 4(x + 7) = 2(x + 12) + 2(x + 1)

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Key Concepts

Types of Equations: Identity, Conditional, and Contradiction

Solving Linear Equations

Solution Sets