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Chapter 4: Problem 32

Graph and write interval notation for each compound inequality. $$ -8

Short Answer

Expert verified
(-8, -2]

Step by step solution

01

Understand the Inequality

The given compound inequality is -8 < x and x less than or equal to -2 . It means any value of x must be greater than -8 and less than or equal to -2.
02

Graph the Inequality

To graph the inequality -8 < x and x 鈮� -2 on a number line: - Draw a number line. Highlight the region between -8 and -2 with -8 being open (indicated by a hollow circle), and -2 being closed (indicated by a filled-in circle).
03

Write Interval Notation

To express this compound inequality in interval notation, use (-8, -2]. This represents all the numbers greater than -8 but less than or equal to -2.

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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 of writing subsets of the real numbers in a compact form. It uses brackets to show which values are included in the set and which are not.
For our given inequality \(-8 < x \leq -2\), we write it in interval notation as \((-8, -2]\).
Here鈥檚 how it works:
  • Parentheses \( ( ) \) indicate that the endpoint is not included in the interval. For example, \( -8 < x \), which means -8 is not part of the solution, so we use \( (\).
  • Brackets \( [ ] \) indicate that the endpoint is included in the interval. For \( x \leq -2 \), -2 is part of the solution, hence we use \[ \].
Combining these, the interval from -8 to -2 where -8 is not included, but -2 is included, is written as \((-8, -2]\).
Number Line Graphing
Graphing inequalities on a number line helps visualize the range of values that satisfy the inequality.
Let鈥檚 go step-by-step to graph \(-8 < x \leq -2\):
  • Draw a horizontal line and mark points for -8 and -2 on it.
  • For \(-8 < x\), use a hollow circle at -8 to indicate that -8 is not part of the solution set.
  • For \(x \leq -2\), use a filled-in circle at -2 to show that -2 is included in the solution set.
  • Shade the line between -8 and -2 to show all numbers greater than -8 and less than or equal to -2.

This shading represents all possible values of x that satisfy both parts of the compound inequality.
Algebraic Inequalities
Algebraic inequalities express a range of possible values for a variable. In our example, \(-8 < x \leq -2\), the inequality tells us that x must be:
  • Greater than -8
  • Less than or equal to -2

To solve such inequalities, we typically:
  • Isolate the variable on one side of the inequality.
  • Apply the same operations to both sides while remembering to reverse the inequality if multiplying or dividing by a negative number.
Compound inequalities, like the one in our example, involve two separate inequalities combined into one statement. They can often be expressed in interval notation for concise representation.

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