Question

Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. -2(x + 3) = -6(x + 7)

Accepted Answer

Start by expanding both sides of the equation to simplify it. Use the distributive property: multiply -2 by each term inside the parentheses on the left side, and -6 by each term inside the parentheses on the right side. This gives you: $$-2 	imes x + (-2) 	imes 3 = -6 	imes x + (-6) 	imes 7. $$Rewrite the expanded equation explicitly: $$-2x - 6 = -6x - 42. $$Next, collect like terms by adding or subtracting terms to isolate the variable on one side. For example, add $$6x to $$both sides and add $$6 to $$both sides to move all variable terms to one side and constants to the other. After simplifying, you will get an equation in the form $$ax = b$$, where $$a$$ and $$b$$ are constants. Analyze this equation: if $$a 
eq 0$$, then solve for $$x by $$dividing both sides by $$a$$; if $$a = 0$$ and $$b = 0$$, the equation is an identity (true for all $$x$$); if $$a = 0$$ and $$b 
eq 0$$, the equation is a contradiction (no solution). Based on the simplified form, determine whether the original equation is an identity, a conditional equation, or a contradiction, and state the solution set accordingly.

Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. -2(x + 3) = -6(x + 7)

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Key Concepts

Types of Equations: Identity, Conditional, and Contradiction

Solving Linear Equations

Solution Sets