You know an equation has all real number solutions when solving it leads to a true statement that is always true, regardless of the value of the variable, often in the form of 0 = 0. This indicates that the equation is an identity.
Here's a breakdown:
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What does it mean to have all real number solutions? It means that any real number you substitute for the variable in the equation will make the equation true.
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How does this manifest when solving? When you simplify the equation through valid algebraic manipulations, the variable terms will cancel out, leaving you with a true statement that doesn't involve the variable.
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Example: Consider the equation: 2(x + 3) = 2x + 6
Let's solve it:
- Distribute the 2 on the left side: 2x + 6 = 2x + 6
- Subtract 2x from both sides: 6 = 6
- Subtract 6 from both sides: 0 = 0
Since we arrived at 0 = 0, which is always true, this equation has all real numbers as solutions. Any value of 'x' you plug in will satisfy the original equation.
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Contrast with a Conditional Equation: A conditional equation is true only for specific values of the variable. For example, x + 2 = 5 is only true when x = 3. Solving it leads to x = 3, a single solution.
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Contrast with a Contradiction: Some equations, when solved, lead to a false statement (e.g., 0 = 1). These equations have no solution.
In summary, when solving an equation, if you reach a point where the variables disappear and you are left with a true statement like 0=0 or 5=5, then the solution set is the set of all real numbers. This signifies that the original equation is an identity, meaning it holds true for any real number substituted for the variable.
