Problem 85 Solve each inequality. Write the... [FREE SOLUTION]

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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

Isolate the absolute value expression

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

Rewrite the inequality without the absolute value

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

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:

Isolate the absolute value term $ |x| $
Rewrite the inequality without absolute value $ |x| < 4 $
Split into two inequalities $ -4 < x < 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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91影视

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

Short Answer

Step by step solution

Isolate the absolute value expression

Rewrite the inequality without the absolute value

Write the solution set in interval notation

Key Concepts

Understanding Absolute Value

Interval Notation Explained

Basics of Algebra

One App. One Place for Learning.

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