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)

Verified step by step guidance

1. 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 \(8x + 28 = 2x + 22 + 6x + 6\).
2. 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 \(8x + 28 = 8x + 28\).
3. 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.
4. 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 \(x\).
5. Therefore, conclude that the equation is an identity, and the solution set is all real numbers, often written as \(\{ x \mid x \in \mathbb{R} \}\).

Key Concepts

Types of Equations: Identity, Conditional, and Contradiction

An identity is an equation true for all values of the variable, a conditional equation is true only for specific values, and a contradiction has no solution. Recognizing these types helps determine the nature of the solution set.

Solving Linear Equations

Solving linear equations involves simplifying both sides, combining like terms, and isolating the variable. This process helps find the values that satisfy the equation or determine if no or infinite solutions exist.

Solution Sets

The solution set is the collection of all values that satisfy the equation. For identities, it includes all real numbers; for conditional equations, specific values; and for contradictions, it is empty.