"Solve for all real solutions" means to find all the values that are real numbers which, when substituted into a given equation or inequality, make the equation or inequality true.
Understanding the Terms
- Solve: To find the value(s) of the variable(s) that satisfy a given equation or inequality.
- Real Solutions: Solutions that are real numbers. A real number is any number that can be plotted on a number line. This includes rational numbers (like integers, fractions, and terminating or repeating decimals) and irrational numbers (like √2 or π). Real numbers exclude imaginary numbers (involving the square root of negative one, denoted as i).
- All: This implies that you must find every real number that satisfies the equation. No solutions should be omitted.
How to Approach "Solve for All Real Solutions"
- Identify the Equation/Inequality: Clearly understand the mathematical expression you need to solve.
- Apply Appropriate Techniques: Use relevant algebraic manipulations, factoring, quadratic formula (if applicable), trigonometric identities, or other mathematical tools to isolate the variable and find potential solutions.
- Check for Real Number Solutions: After finding potential solutions, verify that they are indeed real numbers. Discard any imaginary or complex solutions if the problem specifically asks for real solutions.
- Verify the Solutions: Substitute the real solutions back into the original equation or inequality to ensure they satisfy the expression.
- State All Solutions: Present all the verified real number solutions.
Examples
- Equation: x2 = 4
- Solutions: x = 2, x = -2 (Both are real numbers).
- Equation: x2 + 1 = 0
- Solutions: x = i, x = -i (Both are imaginary numbers, so there are no real solutions).
- Equation: x - 3 = 0
- Solution: x = 3 (A real number).
- Equation: |x| = 2
- Solutions: x = 2, x = -2 (Both are real numbers).
In essence, "solve for all real solutions" restricts the solution set to only include real numbers, excluding complex or imaginary solutions. You are looking for all numbers that can be placed on the number line that make the equation or inequality true.
