When a solution to an equation or inequality is all real numbers, it signifies that any real number, regardless of its value, will satisfy the given condition. In other words, every single real number you can think of, whether it's positive, negative, zero, a fraction, a decimal, or irrational, will make the statement true.
Understanding "All Real Numbers" as a Solution
Real Numbers Defined
First, let's clarify what "real numbers" encompass. Real numbers include:
- Integers: Such as -3, -2, -1, 0, 1, 2, 3...
- Rational Numbers: Fractions or decimals that terminate or repeat, like 1/2, -3/4, 0.75, or 0.333...
- Irrational Numbers: Numbers that cannot be expressed as a simple fraction, like √2, π, or e.
Essentially, every number on the number line is a real number.
Implications of All Real Numbers as a Solution
According to the provided reference, “If an inequality has all real numbers as the solution, this means that every real number can be substituted into the inequality to make a true statement.” This means there are no restrictions on which values will work for the given equation or inequality.
Scenarios Where All Real Numbers Can Be a Solution
-
Tautological Inequalities: These are inequalities that are always true, no matter what value you substitute for the variable. For example, an inequality like x² + 1 > 0 will always be true, since squaring any real number will result in a positive number or zero, and adding 1 always results in a value greater than zero.
-
Equations That Simplify to a True Statement: If, when solving an equation, the variables disappear, and you're left with a true statement (e.g., 5 = 5), the solution is all real numbers. This means that the original equation will be true for any real number value of the variable.
Examples
| Equation/Inequality | Solution | Explanation |
|---|---|---|
| x² + 4 ≥ 0 | All Real Numbers | Squaring any real number results in a positive value or zero. Adding 4 to that makes it always greater than or equal to zero. |
| x - x = 0 | All Real Numbers | The equation x-x will always equal to zero, for any value of x. This always results in a true statement regardless the value of the variable. |
| 2(x + 3) = 2x + 6 | All Real Numbers | Expanding the left side of the equation gives 2x + 6. The variables will disappear and you are left with 6 = 6. |
Key Takeaway
When a solution set is described as all real numbers, it signifies that no matter what real number you choose, it will make the equation or inequality a valid and true statement. There are no restrictions or specific conditions that need to be met by the variable within the set of real numbers.
