A solution set in algebra is the set of all values that satisfy a given equation or inequality. For example, the solution set for the equation 5m = 15 is {3}.
A solution set represents all the possible values for a variable that make a mathematical statement true. Here's a breakdown with examples:
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Definition: A solution set is the collection of all values that, when substituted for the variable(s) in an equation or inequality, make the statement true.
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Example 1: Simple Equation
- Equation: 5m = 15
- Solution: m = 3
- Solution Set: {3} This means only the value 3 makes the equation 5m = 15 true.
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Example 2: Inequality
- Inequality: x + 2 > 5
- Solution: x > 3
- Solution Set: The set of all real numbers greater than 3. This is often written in interval notation as (3, ∞).
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Example 3: Multiple Solutions
- Equation: x2 = 4
- Solutions: x = 2 and x = -2
- Solution Set: {2, -2} Both 2 and -2, when squared, equal 4.
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Example 4: No Solution
- Equation: |x| = -1 (where |x| is the absolute value of x)
- Solution: There is no real number whose absolute value is -1.
- Solution Set: {} or Ø (the empty set).
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Example 5: System of Equations
- Equations:
- x + y = 5
- x - y = 1
- Solution: x = 3, y = 2
- Solution Set: {(3, 2)} This is the only ordered pair that satisfies both equations.
- Equations:
In summary, a solution set is the complete set of answers to an equation or inequality. It can contain a single value, multiple values, an infinite number of values, or no values at all.
