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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 problem. Harold's mother, who lives across the street, is pouring a concrete driveway, 12 feet wide and 4 inches thick, from the street straight to her house. This is too much work for Harold to do in one day, so his mother has agreed to buy 4 cubic yards of concrete each Saturday for three consecutive Saturdays. How far is it from the street to her house?

Solve each equation. $$-3.48 x+6.981=4.329 x-6.851$$

Solve each problem by using a compound inequality. See Example 12. Selling-price range. Renee wants to sell her car through a broker who charges a commission of \(10 \%\) of the selling price. The book value of the car is \(\$ 14,900,\) but Renee still owes \(\$ 13,104\) on it. Although the car is in only fair condition and will not sell for more than the book value, Renee must get enough to at least pay off the loan. What is the range of the selling price?

Show a complete solution to each problem. Really odd integers. If the smaller of two consecutive odd integers is subtracted from twice the larger one, then the result is \(13 .\) Find the integers.

Solve each problem. The area of a rectangular floor is 23 square yards. The width is 4 yards. Find the length.
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