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Chapter 2: Problem 102

Graph each inequality, and write it using interval notation. \(-3 \leq x<0\)

Short Answer

Expert verified
The interval notation is \([-3, 0)\).

Step by step solution

01

- Identify the inequality

The inequality provided is \(-3 \leq x<0\). This means that the variable x is between -3 and 0, inclusive of -3 and exclusive of 0.
02

- Interpret the inequality

The inequality \(-3 \leq x<0\) can be broken down as follows: \(-3 \leq x\) means x is greater than or equal to -3, and \(x<0\) means x is less than 0.
03

- Write the interval notation

The interval notation for the inequality \(-3 \leq x<0\) is \([-3, 0)\). The bracket [ indicates that -3 is included in the interval, and the parenthesis ) indicates that 0 is not included in the interval.
04

- Graph the inequality

On a number line, mark a solid dot at -3 to indicate that -3 is included in the interval. Draw a line from -3 to 0. Mark an open circle at 0 to indicate that 0 is not included in the interval.

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Key Concepts

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

Interval Notation
Interval notation is a way to represent a range of values on a number line. It makes it easier to see which numbers are included or excluded.

When using interval notation, we use brackets \(\left[ \right]\) to include a number and parentheses \(\left( \right)\) to exclude a number. For example:
  • \([a, b]\): a and b are both included in the interval.
  • \((a, b)\): a and b are both excluded from the interval.
For the inequality \(-3 \leq x < 0\), we use \([\) for \text{-3 is included and \()\) for 0 because 0 is not included. So, the interval notation is \([-3, 0)\).
Inequality Solutions
Inequality solutions show the set of values that satisfy the given inequality.

For \(-3 \leq x < 0\), we need to determine which values x can take:
  • \(-3 \leq x\) means x can be -3 or more.
  • \(x < 0\) means x can be less than 0.
Putting these together, x can be any value between -3 and 0. However, x will include -3 but not 0. Visualization helps!

Think of x values as points or places on a number line. If -3 is included, it means our solution area starts from -3. Note that 0is not included.
Number Line Representation
Graphing inequalities visually shows the range of possible values.

For \(-3 \leq x < 0\), we draw this on a number line:
  • Mark -3 with a solid dot since it is included.
  • Mark 0 with an open circle because 0 is not included.
  • Draw a line between the dots to show the values in this range.
The solid dot at -3 means -3 is included. The open circle at 0 means 0 isn't. The line between shows everything in between the dots.

This way, anyone can look at the number line and see what values work for the inequality.

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