Dealing with absolute values in algebra can be particularly challenging for several reasons.

While the basic concept of absolute values may be easy to understand, managing inequalities and equations requires a deeper understanding and greater interest in the subject which is where an absolute value inequalities calculator can be helpful.

An absolute value inequality features an absolute value expression with variables. These types of inequalities consist of absolute value functions and inequality symbols and can take various forms. An absolute value inequality is a simple linear expression in one variable with symbols such as >, <, >, <.

For example, an absolute value inequality may be written in one of the following forms or can be converted to one of these forms for easier analysis:

- \( ax + b < c \)
- \( ax + b > c \)
- \( ax + b < c \)
- \(ax + b > c\)

There are four cases to consider when using formulas to solve these inequalities. It is important to note that we assume that “a” represents a positive real number in all cases.

You can solve absolute value inequalities with the help of an absolute value inequalities calculator and an understanding of the formulas involved.

**Case 1: **\(|x| < a\) or \(|x| ≤ a\)

In this scenario, we use the following formulas to determine the solution:

if \( |x| < a\) => \( -a < x < a \)

if \( |x| ≤ a\) => \( -a ≤ x ≤ a \)

**Case 2: **\(|x| > a\) or \(|x| ≥ a\)

To determine the solution in this scenario, we use the following formulas:

if \( |x| > a\) => \( x < -a \) or \( x > a \)

if \( |x| ≥ a\) => \( x ≤ -a \) or \( x ≥ a \)

**Case 3: **\(|x| < -a\) or \(|x| ≤ -a\)

Let’s consider the inequality \(|x| < -a\) (or \(|x| ≤ -a\)). Here, we’re saying that the absolute value of \(x\) is less than (or less than or equal to) some negative number \(-a\).

But wait a minute — isn’t the absolute value always positive? That’s right! So if \(-a\) is negative (which it is, since we’ve assumed that a is a positive real number), then \(|x|\) can never be less than (or less than or equal to) \(-a\).

Put another way, we have a “positive number that is less than (or less than or equal to) a negative number,” which is always false. It’s like saying, “I have $10 in my pocket, but I owe $20.” That’s impossible!

So, because of this contradiction, we can say that there is no solution to the inequality \(|x| < -a\) (or \(|x| ≤ -a\)). There’s no value of \(x\) that could possibly make that inequality true.

**Case 4: **\(|x| > -a \) or \( |x| ≥ -a \)

Now, let’s consider the inequality \(|x| > -a\) (or \(|x| ≥ -a\)). Here, the absolute value of \(x\) is greater than (or greater than or equal to) some negative number \(-a\).

But wait a minute — isn’t the absolute value always positive? That’s right! So if \(-a\) is negative (which it is, since we’ve assumed that a is a positive real number), then \(|x|\) must be greater than (or greater than or equal to) \(-a\), and therefore positive.

So, we have a “positive number that is greater than (or greater than or equal to) a negative number,” which is always true. It’s like saying, “I have $20 in my pocket, and I owe $10.” That’s possible!

Therefore, the solution to the inequality \(|x| > -a\) (or \(|x| ≥ -a\)) is the set of all real numbers, denoted by R. That means any real number can solve this inequality.

To summarize, we can express the solution to the inequality \(|x| > -a\) or \(|x| ≥ -a\) as the set of all real numbers, R.

A grocery store sells apples in a package labelled 2.5 pounds, allowing for a margin of error of 0.3 pounds. Write an absolute value inequality that models this relationship. Then, using your inequality, find the allowable weight range for the apple package that meets these specifications.

**Solution:**

Let us assume that the actual weight of the apple package is x pounds. Then the absolute value inequality that corresponds to the above scenario is,

\(|x – 2.5| ≤ 0.3\)

Using the formulas, we learned:

\(-0.3 ≤ x – 2.5 ≤ 0.3\)

Adding \(2.5\) on all the sides

\(2.2 ≤ x ≤ 2.8\)

The weight of the apple package can be in the range of [2.2, 2.8] pounds.

How does absolute value impact inequalities?

Absolute value significantly impacts inequalities by transforming a simple inequality into a compound inequality. Therefore, isolating the absolute value expression on one side of the inequality is essential to accurately solve the equation. For example, if the absolute value expression equals a negative number, the absolute value equation has no solution, as absolute value can never be negative.

Do absolute value inequalities always have two solutions?

An absolute value inequality may or may not have two solutions, depending on the given expression. Sometimes, an absolute value inequality may have a single or no solution.

Is it possible for an absolute value inequality to be less than a negative number?

No, this is because the absolute value of any number is always non-negative; it can never be less than zero. Therefore, an absolute value inequality involving a variable cannot have a solution that is less than or equal to a negative number.

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