solution set
the set of all possible solutions
A solution set refers to the set of values that satisfy a given equation or inequality. For instance, the solution set of the equation x + 2 = 5 is {3}, meaning that the value of x that satisfies the equation is 3. Likewise, the solution set of the inequality 4x – 3 ≤ 9 is {x ≤ 3}, indicating that x can take any value less than or equal to 3 to satisfy the inequality.
A solution set can be either finite or infinite depending on the nature of the equation or inequality. For example, the quadratic equation x² + 3x + 2 = 0 has two distinct solutions, and hence its solution set is finite, {x = -1, -2}. On the other hand, the inequality x² – 4 < 0 has an infinite solution set of {x ∈ (-2, 2)}, which means that any value of x between -2 and 2, not including the endpoints, will satisfy the inequality. It is important to note that some equations or inequalities can have no solution at all. For instance, the equation x² + 1 = 0 has no real solutions because no real number squared can be equal to a negative number. In such cases, we say that the solution set is the empty set, denoted by { }.
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