Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-12: Express the inequality as an interval, and sketch its graph. $$ -3 \geq x>-5 $$

Step-by-Step Solution

Answer

The interval is \((-5, -3]\).

Step 1: Identify the inequality constraints

The inequality given is \(-3 \geq x > -5\). This can be broken down into two parts: \(-3 \geq x\) and \(x > -5\).

Step 2: Interpret the inequality on the number line

The inequality \(-3 \geq x\) means that \(x\) is less than or equal to \(-3\), which includes \(-3\). The inequality \(x > -5\) means \(x\) is greater than \(-5\) but does not include \(-5\).

Step 3: Express the solution as an interval

Combining the two parts, \(x\) must be greater than \(-5\) and less than or equal to \(-3\). In interval notation, this is expressed as \((-5, -3]\).

Step 4: Sketch the graph of the inequality

Draw a number line and shade the region between \(-5\) and \(-3\). Place an open circle at \(-5\) to indicate that \(-5\) is not included in the interval, and a closed dot at \(-3\) to show that \(-3\) is included.

Key Concepts

Interval Notation

Interval notation is a special way of writing the range of values that a variable can take in an inequality. It uses brackets and parentheses to show whether endpoints are included or excluded.
For example:

• Brackets, such as \([ , ]\), indicate that the number is included in the interval. For example, \(-3\) is included in \([-3, x]\).


• Parentheses, such as \(( , )\), indicate that the number is not included. For example, \(-5\) is not included in \((-5, x]\).

In the exercise, the interval notation was \((-5, -3]\), which means:

• \(x\) is greater than \(-5\) (but \(-5\) is not included).


• \(x\) is less than or equal to \(-3\) (and \(-3\) is included).

Number Line

A number line is a line that visually represents numbers at evenly spaced intervals. It's a great tool for understanding inequalities as it lets you see how numbers relate to each other in terms of greater or lesser values.
When working with inequalities, a number line helps to picture where values fall between other values:

• Numbers to the left are always smaller.
• Numbers to the right are always larger.

To depict an inequality like \(-3 \geq x > -5\), mark the numbers \(-3\) and \(-5\) on the line.
Shade the part between them to show all possible values \(x\) can take.
An open circle on \(-5\) shows this number isn't included.
A closed circle or filled dot on \(-3\) shows that this number is included in the values \(x\) can take.

Graphing Inequalities

Graphing inequalities on a number line visually communicates the solutions of inequalities. This method allows you to easily see which values \(x\) can be, based on the constraints given.
Here's how you can graph inequalities like the one in the exercise:

• Identify the critical points (where the inequality changes).


• Use open circles for values that are not included (as in \(x > -5\)).


• Use closed dots for values that are included (as in \(-3 \geq x\)).


• Shade the section of the number line where the inequality holds true.

In the given inequality \(-3 \geq x > -5\), mark an open circle on \(-5\) and a closed dot on \(-3\). Then, shade the region between these points. This shaded area represents all values that \(x\) can take, making it simpler to understand and discuss the inequality.