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Chapter 2: Problem 85

Solve each inequality. Write the solution set using interval notation. $$5>|x|+1$$

Short Answer

Expert verified
(-4, 4)

Step by step solution

01

Isolate the absolute value expression

Subtract 1 from both sides of the inequality to isolate the absolute value expression. 5 - 1 > |x| 4 > |x|
02

Rewrite the inequality without the absolute value

The inequality |x| < 4 can be rewritten as two separate inequalities: -4 < x < 4
03

Write the solution set in interval notation

The solution set for the inequality -4 < x < 4 can be expressed in interval notation as: (-4, 4)

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Key Concepts

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

Understanding Absolute Value
Absolute value measures the distance of a number from zero on a number line. It is always non-negative. The absolute value of x is written as \( |x| \). Whether x is positive or negative, \( |x| \) gives the same result. For example:
  • \( |3| = 3 \)
  • \( |-3| = 3 \)
In the given problem \( 5 > |x| + 1 \), we need to isolate \( |x| \). Subtracting 1 from both sides, we get \( 4 > |x| \). This tells us that the distance of x from 0 should be less than 4 units.
Interval Notation Explained
Interval notation is a way of writing subsets of the real number line. It simplifies expression of the range within which a number lies. Consider the interval \( (-4, 4) \). This means all numbers greater than \(-4\) and less than \(4\).
Here's a breakdown of interval notation:
  • Parentheses \( () \): The endpoints are not included (open interval)
  • Brackets \( [] \): The endpoints are included (closed interval)
  • Combining: \( (a, b) \) means \(a < x < b\) and \( [a, b] \) means \(a \leq x \leq b\)
So, solving \( 4 > |x| \) gives us \( -4 < x < 4 \). In interval notation, this is \( (-4, 4) \).
Basics of Algebra
Algebra involves working with variables, numbers, and operations to solve for the unknown. In our problem, the steps to solve the inequality:
  1. Isolate the absolute value term \( |x| \)
  2. Rewrite the inequality without absolute value \( |x| < 4 \)
  3. Split into two inequalities \( -4 < x < 4 \)
  4. Express the solution using interval notation \( (-4, 4) \)
Always remember: algebraic manipulation like adding, subtracting, multiplying, or dividing helps to isolate the variable. Breaking down absolute value equations into two separate inequalities is often a crucial step.

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Most popular questions from this chapter

Solve each equation. Practice combining some steps. Look for more efficient ways to solve each equation. $$-3-z=3$$

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