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Chapter 4: Problem 46

Graph and write interval notation for each compound inequality. $$ 6>-x \geq 0 $$

Short Answer

Expert verified
Interval notation: \((-6, 0]\).

Step by step solution

01

- Isolate variable

First, we need to rewrite the compound inequality in a way that makes it easier to understand. The inequality given is: \(6 > -x \geq 0\). To isolate \(-x\) and make it positive, we need to multiply every part of the inequality by -1. Remember, when you multiply or divide an inequality by a negative number, the direction of the inequality signs must be reversed. Therefore, multiplying by -1 we get: \(-6 < x \leq 0\)
02

- Write in Interval Notation

Now, convert the inequality \(-6 < x \leq 0\) into interval notation. This notation represents the set of solutions for the inequality. The interval notation combines the lower bound and the upper bound of the inequalities: \((-6, 0]\)
03

- Graph the Interval

To graph the interval \((-6, 0]\), draw a number line. Plot a point at -6 with an open circle to represent that -6 is not included. Plot another point at 0 with a filled circle to indicate that 0 is included. Draw a line connecting the open circle to the filled circle to show all the values between -6 and 0 are part of the solution.

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Key Concepts

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

Isolating Variables
Isolating variables is a critical step in solving inequalities. In our example, we start with the compound inequality: \(6 > -x \, \geq \, 0\). To isolate \(x\), we need to reverse the inequalities by multiplying every term by -1. Remember, multiplying or dividing an inequality by a negative number flips the inequality signs.

Hence: \( 6 > -x \, \geq \, 0 \) becomes \( -6 < x \, \leq \, 0 \). This step ensures that \(x\) is positive and easier to work with.
Interval Notation
Interval notation is a way to represent the set of solutions for inequalities. Once the inequality \( -6 < x \, \leq \, 0 \) is obtained, converting it into interval notation is straightforward.

The inequality says that \(x\) is greater than -6 but less than or equal to 0. In interval notation, this is written as \((-6, 0]\).

In this notation:
  • The parenthesis \(( \) around -6 indicates that -6 is not included in the solution.
  • The bracket \([ \) around 0 indicates that 0 is included in the solution.
This concise form quickly shows the range of possible values.
Graphing Inequalities
Graphing inequalities is a visual way to understand the range of solutions. For the interval \((-6, 0]\), follow these steps to graph it on a number line:
  • Draw a straight horizontal line representing the number line.
  • Mark -6 with an open circle to indicate -6 is not included in the interval.
  • Mark 0 with a filled circle since 0 is included in the interval.
  • Draw a line connecting the open circle at -6 to the filled circle at 0.
This line shows all the values between -6 and 0 that satisfy the inequality, providing a clear visual representation.

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