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Chapter 2: Problem 20

Solve inequality. Write the solution set interval notation, and graph it. \(p-3 \geq-11\)

Short Answer

Expert verified
The solution set is \( [-8, \infty) \), and it is represented with a closed circle at \( -8 \) shading to the right on the number line.

Step by step solution

01

Isolate the variable

To solve for the variable, add 3 to both sides of the inequality: \( p - 3 + 3 \geq -11 + 3 \) This simplifies to: \( p \geq -8 \)
02

Write the solution in interval notation

Since \( p \) is greater than or equal to \( -8 \), the interval notation is: \( [-8, \infty) \)
03

Graph the solution set

To graph the solution set, draw a number line. Place a closed circle at \( -8 \) to indicate that \( -8 \) is included in the solution set. Shade the number line to the right of \( -8 \) to represent all numbers greater than or equal to \( -8 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate the Variable
To solve inequalities, we first need to isolate the variable. This means getting the variable by itself on one side of the inequality sign. Let's take an example to understand this better. Consider the inequality:

\( p - 3 \geq -11 \)

The goal is to have \( p \) on one side. To do this, we need to get rid of the \'-3\' that's being subtracted from \( p \).

We do this by adding 3 to both sides of the inequality:

\( p - 3 + 3 \geq -11 + 3 \)

Adding 3 cancels out the \'-3\' on the left side, simplifying our inequality to:

\( p \geq -8 \)

Now, \( p \) is isolated, and we can proceed to the next steps.
Interval Notation
Once we have isolated the variable, the next step is to express the solution using interval notation. Interval notation is a way of writing subsets of the real number line. It is very handy for expressing solutions to inequalities.

Let's continue with our example:

  • We have determined \( p \geq -8 \).

  • This means \( p \) can be any number from \( -8 \) to infinity, including \( -8 \) itself.
In interval notation, this is written as:

\( [-8, \infty) \)

Here鈥檚 how to read it:

  • The square bracket \( [ \) means \( -8 \) is included in the interval (it satisfies the inequality \( p \geq -8 \)).
  • The parenthesis \( ) \) after infinity means infinity is never included鈥攊t鈥檚 a concept, not a number.
This compact notation clearly shows the range of values that satisfy the inequality.
Graphing Solution Sets
Visualizing the solution set on a number line can make understanding inequalities much easier. This is how you can graph the solution set:

  1. Draw a horizontal line and mark it with numbers.

  2. Identify the key point鈥攊n our example, \( -8 \).
  3. Place a closed circle at \( -8 \) because \( p \geq -8 \) includes \( -8 \).
  4. Shade the number line to the right of \( -8 \) to show all numbers greater than or equal to \( -8 \).
This shaded area represents all possible values of \( p \) that satisfy the inequality. Here's what it looks like:

A closed circle at \( -8 \) and a shaded arrow extending to the right towards infinity.

By drawing this graph, we can easily see which numbers are included in the solution set.

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