Problem 50 - Precalculus Mathematics for Calculus

Question

Express the interval in terms of inequalities, and then graph the interval. $$(-\infty, 1)$$

Step-by-Step Solution

Answer

The interval \((-\infty, 1)\) as an inequality is \(x < 1\).

Step 1: Understand the Interval

The given interval is \((-\infty, 1)\). This indicates all real numbers less than 1, but not including 1 itself. The interval starts from negative infinity and goes up to, but does not include, the number 1.

Step 2: Express as an Inequality

To express \((-\infty, 1)\) as an inequality, identify the variable that represents the numbers in this interval. Let's use \(x\). Therefore, the inequality representation is \(x < 1\). This means \(x\) can be any number less than 1.

Step 3: Graph the Inequality

To graph the inequality \(x < 1\), represent it on a number line. Plot an open circle at 1 to show that 1 is not included in the interval, and shade all numbers to the left of 1 extending indefinitely. This shading represents all numbers less than 1.

Key Concepts

Interval Notation

Interval notation is a way of representing a set of numbers along an interval on the real number line. It provides a concise way to express where a number lies relative to other numbers. When we see something like \((-\infty, 1)\), it tells us that we are considering all numbers less than 1.
The use of parentheses in interval notation is significant:

• "(" or ")" means the endpoint is not included in the interval.
• "[" or "]" would mean the endpoint is included in the interval.

In the example \((-\infty, 1)\), it features an opening parenthesis "(" next to -∞ which shows the interval starts below any real number, and also a parenthesis ")" next to 1 indicating that 1 itself is not part of the interval. Interval notation is efficient for writing mathematical solutions since it simplifies the expression of ranges of numbers.

Number Line

A number line is a visual tool that helps us understand the set of numbers and their relative positions. Imagine a straight line where numbers are laid out from smallest on the left to greatest on the right.

In contexts involving inequalities or intervals, number lines become particularly helpful. The task involves using a number line to represent numbers less than a given point by shading or marking portions of the line. For \( x < 1 \), which relates to the interval \((-\infty, 1)\), you start marking from somewhere on the far left (indicating -∞) and go up to right before 1, without including 1 itself. An open circle or dot is commonly used at the endpoint 1 to signify that 1 is not in this set. This kind of representation helps in understanding both the directional aspect and the continuity of the set of numbers described by the inequality.

Graphing Inequalities

Graphing inequalities on a number line is a practical technique to visualize solutions. This process involves a few steps to ensure clarity:

• First, determine if the inequality is strict (using < or >) or inclusive (using ≤ or ≥). This affects whether you will use an open or closed circle.
• For the inequality \(x < 1\), you use an open circle at 1, signifying that this point is not part of the solution set.

Once you've marked the endpoint, shade the portion of the number line that includes all values satisfying the inequality. This shading effectively communicates the range of numbers you're accounting for. For \(x < 1\), draw an arrow or shaded line extending from the open circle at 1 all the way to the left, representing all numbers leading up to, but not including, 1.
This visualization helps in both simplifying solutions and confirming your understanding of what the inequality describes.