Question

Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 4(2x + 7) = 2x + 22 + 3(2x + 2)

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 multiply 3 by each term inside the parentheses on the right side. This gives you: $$4(2x + 7) = 2x + 22 + 3(2x + 2)$$ becomes $$8x + 28 = 2x + 22 + 6x + 6. $$Next, combine like terms on the right side of the equation. Add the terms involving $$x$$ and the constant terms separately: $$2x + 6x = 8x$$ and $$22 + 6 = 28. So $$the equation now looks like $$8x + 28 = 8x + 28. $$Now, analyze the simplified equation. Since both sides are identical expressions, this suggests the equation might be an identity. To confirm, subtract $$8x + 28$$ from both sides to see if the equation reduces to a true statement. After subtracting, you get $$0 = 0$$, which is always true regardless of the value of $$x. $$This means the original equation holds for all real numbers $$x. $$Therefore, conclude that the equation is an identity, and the solution set is all real numbers, often written as $$\{ x \mid x \in \mathbb{R} \}.$$

Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 4(2x + 7) = 2x + 22 + 3(2x + 2)

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Key Concepts

Types of Equations: Identity, Conditional, and Contradiction

Solving Linear Equations

Solution Sets