Problem 36 - Algebra and Trigonometry

Question

\(33-38\) . Express the interval in terms of inequalities, and then graph the interval. $$ \left[-6,-\frac{1}{2}\right] $$

Step-by-Step Solution

Answer

The interval is \(-6 \leq x \leq -\frac{1}{2}\), shown on a number line with endpoints -6 and -\(\frac{1}{2}\) closed and shaded between.

Step 1: Understand the Interval

Given the interval \([-6, -\frac{1}{2}]\), this is a closed interval which includes both endpoints. This means that the values include both -6 and -\(\frac{1}{2}\) and every number between them.

Step 2: Write the Inequality

Express the interval \([-6, -\frac{1}{2}]\) as an inequality. This can be represented as \(-6 \leq x \leq -\frac{1}{2}\). This means \(x\) can take any value between -6 and -\(\frac{1}{2}\), inclusive.

Step 3: Graph the Interval

To graph the interval \([-6, -\frac{1}{2}]\), draw a number line. Mark and label the endpoints -6 and -\(\frac{1}{2}\). Since both are included, use a closed (or shaded) dot on both points. Then, shade the entire region between these two points on the number line to represent all the numbers \(x\) where \(-6 \leq x \leq -\frac{1}{2}\).

Key Concepts

Understanding Inequalities

Inequalities are mathematical statements that relate expressions in a non-equal way. They express a range of possible values for a variable. The primary symbols used in inequalities are:

• \(<\) and \(>\) which denote less than and greater than, respectively
• \(\leq\) and \(\geq\) which mean "less than or equal to" and "greater than or equal to"

In the interval \([-6, -\frac{1}{2}]\), the inequality can be written as \(-6 \leq x \leq -\frac{1}{2}\).This includes all values of \(x\) in the specified range, meaning \(x\) is greater than or equal to \(-6\) and less than or equal to \(-\frac{1}{2}\).Here, the brackets indicate that the endpoints are part of the solution set, emphasizing that the range is inclusive.

Visualizing Inequalities on a Number Line

A number line is a visual representation of numbers on a straight line. It helps in understanding the size and position of different numbers relative to each other. When showcasing inequalities, a number line can be incredibly helpful. Imagine our number line displays only a relevant segment that includes \(-6\) to \(-\frac{1}{2}\).- **Start by marking the essential points**: In our example, you would place points at \(-6\) and \(-\frac{1}{2}\).- **Understanding placement**: As the inequality \(-6 \leq x \leq -\frac{1}{2}\), the number line should start shading from \(-6\) and continue to \(-\frac{1}{2}\).- **Use closed dots** to show inclusion of endpoints, indicating these numbers are part of the set solutions.

Defining a Closed Interval

A closed interval is a segment of the real number line which contains both of its endpoints. This characteristic is shown using square brackets, like \([-6, -\frac{1}{2}]\). It contrasts with an open interval, which does not include its endpoints and uses parentheses.When discussing a closed interval:

• The square bracket on either side signifies inclusion, telling us that the values at those points are part of the interval.
• In our interval \([-6, -\frac{1}{2}]\), both \(-6\) and \(-\frac{1}{2} \) are included.

Closed intervals are important because they define a precise range in which a certain solution or set of solutions must fall, always including the borders.

Graphing Intervals Efficiently

Graphing intervals involves representing them visually on a number line. This helps in easily interpreting complex inequality expressions.Here's how you can graph the interval \([-6, -\frac{1}{2}]\):

• Start by drawing a horizontal line, which acts as your number line.
• Locate the points \(-6\) and \(-\frac{1}{2}\) on the line.
• Since this is a closed interval, mark both points with shaded dots.
• Shade the entire span between these dots to indicate all possible values of \(x\) that lie within these limits.

Such graphing not only makes it easier to understand inequalities but also aids in visual learning, giving a clear depiction of the range covered by the interval.