Convert to Interval Notation x< 7 or x< 8
The question is asking to represent the inequality expressions "x < 7" or "x < 8" using interval notation, which is a way of describing the set of numbers that satisfy the inequalities. Interval notation is typically written with brackets or parentheses to denote whether endpoints are included or excluded in the set. Here, you would be required to interpret the two inequalities and express them as a single interval that satisfies the conditions given by the “or” logical connector, indicating that the solution set includes all numbers that satisfy either one of the inequalities or both.
$x < 7$or$x < 8$
Answer
Solution:
Step 1:
Identify the combined set of numbers that satisfy either of the inequalities $x < 7$ or $x < 8$.
Step 2:
Express the solution set in interval notation, which includes all numbers less than 8. $\left(-\infty, 8\right)$
Step 3:
Since there are no additional constraints, the interval notation from Step 2 is the final answer.
Knowledge Notes:
The problem involves understanding inequality and interval notation concepts.
Inequality: An inequality is a mathematical statement that relates two values, indicating that one is greater than, less than, or possibly equal to the other. In this case, $x < 7$ and $x < 8$ are two separate inequalities.
Interval Notation: Interval notation is a way of writing subsets of the real number line. An interval includes all numbers between two endpoints. For example, $(a, b)$ represents all numbers greater than $a$ and less than $b$. Square brackets, $[$ or $]$, are used for inclusive boundaries, while parentheses, $($ or $)$, are used for exclusive boundaries.
Union of Intervals: The union of two intervals includes all numbers that are in either interval. In logical terms, it corresponds to the "or" operation. If we have $x < 7$ or $x < 8$, the union would be all $x$ that satisfy either condition.
Infinite Intervals: When an interval extends indefinitely in one direction, it is represented by infinity ($\infty$) or negative infinity ($-\infty$). In interval notation, these are always written with parentheses to indicate that infinity is not a number that can be reached.
In this problem, since $x < 8$ includes all numbers that are also in $x < 7$, the interval notation is $\left(-\infty, 8\right)$, which represents all real numbers less than 8.
