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Chapter 6: Problem 15

Solve the inequality. Then graph and check the solution. $$ |x|<15 $$

Short Answer

Expert verified
The solution is that x could be any real number between -15 and 15 and is represented as (-15, 15) in interval notation.

Step by step solution

01

Breakdown of the Absolute Value Inequality

The absolute value of a number is its distance from zero, so \(|x|<15\) tells us that x is less than 15 units away from zero in both directions. This can be represented as -15 < x < 15.
02

Write the Solution as an Inequality

Inequality can be written as : -15 < x < 15, which means x is greater than -15 and less than 15.
03

Graph and Check the Solution

Now, plot -15 and 15 on a number line. All the numbers between -15 and 15 are the solution to the inequality. The interval will be open because x is not equal to -15 and 15.

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Key Concepts

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

Absolute Value
Absolute value refers to the magnitude of a number without considering its sign. It's essentially how far the number is from zero on a number line. This is always expressed as a positive number or zero. For example, the absolute value of both -7 and 7 is 7.
  • Denoted by two vertical bars: for a number \(a\), it's written as \(|a|\).
  • Mathematically, \(|x| < 15\) means x is within 15 units from zero on the number line, both to the left and right.
  • This inequality expands to -15 < x < 15, covering a range of numbers that are neither -15 nor 15.
Understanding absolute value as a measure of distance helps us grasp its application in solving inequalities.
Number Line Graphing
Graphing on a number line gives us a visual representation of solutions to inequalities. It helps identify what numbers satisfy the given condition. Let's see how we graph \(-15 < x < 15\):
  • Draw a horizontal line to represent the number line.
  • Mark the critical points, -15 and 15, on this line.
  • Because the inequality is strict (x is neither -15 nor 15), use open circles on -15 and 15 to signify they are not included.
  • Shade or draw a line segment between -15 and 15, indicating all numbers x between these values satisfy the inequality.
This graphical method makes it easier for us to understand which values meet the conditions of the inequality.
Interval Notation
Interval notation is a concise way to express a range of numbers that satisfy an inequality. It provides a clearer, mathematical representation of the solution set.
  • The inequality \(-15 < x < 15\) translates to interval notation as \((-15, 15)\).
  • Use parentheses, \(()\), to indicate that end values -15 and 15 are not part of the solution (open interval).
  • If the inequality included the end values, square brackets, \([])\), would be used, indicating a closed interval.
Using interval notation streamlines the way we describe solution sets and ensures clarity, especially when dealing with complex inequalities.

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