Absolute value equations and inequalities are fundamental tools for solving problems involving non-negative values, such as distance. For instance, imagine visiting a friend who lives on the 4th floor of a 10-floor building. Unsure which floor they reside on, you call and ask. Your friend responds that they're on the 4th floor and you respond by saying you are two floors away. But which floor are you on? The 2nd or 6th? Either could be two floors away from the 4th floor, making the distance unchanged regardless of going up or down.

Absolute value inequalities are also useful for calculating margins of error or tolerance, making them valuable in fields like manufacturing where accurate measurements of weight, length, and temperature are crucial.

Absolute value is the positive magnitude of a number, denoted as |x|. Why is this? Absolute value measures the distance from zero to a specific number on the number line. For example, the distance from zero to 2 is 2, and the distance from zero to -2 is also 2. Thus, |2| and |-2| both equal 2.

This is why |x| represents the value of a number no matter its sign. When there is an expression inside the absolute value, the value is calculated, and the positive version of the result is taken.

**Example:** Evaluate |x-3|+1 if x=-2

|x-3|+1 = |-2-3|+1 = |-5|+1 = 5 + 1 = 6

The absolute value of any real number x is represented as:

**|x| = x, if x ≥ 0**

**|x| = -x, if x < 0**

From these expressions, it is clear that if the number inside the absolute value is already positive, it remains unchanged. However, if the number is negative, the result is the positive version of that number, as if the negative number is multiplied by -1.

The properties of absolute value are:

- The absolute value of a number always results in a positive number.
- |4| = 4 and |-5| = 5
- The absolute value of a number x is the same as the absolute value of -x.
- |x| = |-x| = x
- The absolute value of the product of two real numbers can be separated into two individual absolute values.
- |a × b| = |a| × |b|
- |2(-3)| = 2 × |-3| = 2 × 3 = 6
- The absolute value of the division of two numbers can also be separated into two individual absolute values.
- ab = |a| ÷ |b|
- -9 ÷ 3 = |-9| ÷ |3| = 9 ÷ 3 = 3
- The absolute value of the sum or difference of two numbers cannot be separated into the sum or difference of two individual absolute values.
- |a ± b| ≠ |a| ± |b|

Absolute value equations include absolute value expressions and can be solved using the following steps:

- Determine the solution when the value inside the absolute value is positive (a=b).
- Determine the solution when the value inside the absolute value is negative (-a=b).
- Check each solution by substituting its value into the original equation to see if it remains true.
- Define the solution set.
- If needed, plot the solutions on the number line.

**Example:** Solve |x-5| = 4

**1. Solution 1 (a=b):**

|x-5| = 4

x-5 = 4

x = 9

**2. Solution 2 (-a=b):**

-(x-5) = 4

-x+5 = 4

-x = -1

x = 1

**3. Check the solutions:**

**a) Check Solution 1 (x=9):**

|x-5| = 4

|9-5| = 4

|4| = 4

4 = 4 **✔**

**b) Check Solution 2 (x=1):**

|x-5| = 4

|1-5| = 4

|-4| = 4

4 = 4 **✔**

**4. Plot on Number Line:**

If we represent the solutions on the number line, it becomes evident that both values are four units away from five.

In mathematics, absolute value equations and inequalities involve absolute value expressions, represented by a single vertical bar enclosing a number. Absolute value represents the distance of a number from zero on a number line, making it a crucial concept in algebra.

To solve an absolute value equation, there are three essential steps to follow:

- 1. Determine the solution when the value within the absolute value bars is positive.
- 2. Find the solution when the value within the absolute value bars is negative.
- 3. Verify the solutions by substituting them into the original equation.

It is crucial to consider both positive and negative solutions, as the value within the absolute value can be either positive or negative.

The process for solving absolute value inequalities is similar to that of absolute value equations. However, we must also consider the possibility of two solutions.

For all real numbers a and b, where b ≥ 0:

- If |a| < b, then -b < a < b
- If |a| > b, then a > b or a < -b

To solve an absolute value inequality, rewrite it as a compound inequality and combine them using the words "and" or "or". Then, define the solution set and graph it on a number line.

- Absolute value equations and inequalities involve absolute value expressions.
- When solving equations, consider both positive and negative solutions for the value within the absolute value.
- Absolute value inequalities can be solved by rewriting them as compound inequalities.
- The solution of an inequality can be represented on a number line, using an empty circle to represent that x is not part of the solution, and a closed circle if x is part of the solution.

By keeping these key points in mind, you can confidently solve absolute value equations and inequalities in your math studies.

Example 1: Solve |2x-5| > 9

We can rewrite this equation as a compound inequality: -9 < 2x-5 < 9

Solving for each case separately:

- -9 < 2x-5
- 2x > -4
- x > -2

- 2x-5 < 9
- 2x < 14
- x < 7

Therefore, the solution set is x|-2 < x < 7

The solution can be graphed on a number line with an open circle at -2 and a closed circle at 7.

Example 2: Solve |3x+1| < 11

This can be rewritten as: -11 < 3x+1 < 11

Solving for each case separately:

- -11 < 3x+1
- 3x > -12
- x > -4

- 3x+1 < 11
- 3x < 10
- x < 3.33

The solution set is x|-4 < x < 3.33

The solution can be graphed on a number line with an open circle at -4 and a closed circle at 3.33.

In algebra, an absolute value equation is a mathematical expression containing a variable within absolute value bars that evaluates to a positive value regardless of the sign of the contents. To solve such equations, follow these three key steps to find the appropriate solution set.

- 1. Identify the Positive Case (a = b)
- 2. Consider the Negative Case (-a = b)
- 3. Verify Solutions by Checking the Original Equation

First, consider the positive case where the absolute value expression is equal to a positive value. This means that the variable within the absolute value bars must be equal to the positive value, so a = b. In this case, the solution is simply the value of the variable.

Next, consider the negative case where the absolute value expression is equal to a negative value. This can be solved by rewriting the equation as a compound inequality and finding two solutions, one for when the value within the absolute value is greater than the negative value and one for when it is less than the negative value.

Finally, verify all solutions by substituting them into the original equation to ensure they remain true. In combination, these three steps will lead to the accurate solution set for any absolute value equation.

Solving an absolute value equation involves finding the possible values of the variable within the absolute value bars in order to satisfy the equation. This process requires three important steps: identifying the positive and negative cases, finding the solutions, and verifying their accuracy.

In the case of -a = b, the variable within the absolute value bars must be equal to the negative value. This means that the solution to the equation is the negative value of the variable, a = -b. This holds true for any absolute value equation where the variable must be equal to a negative value.

To find the solutions, we must isolate the variable by multiplying both sides of the equation by -1. This results in a = -b, and we can then substitute the value of b into the equation to determine the value of a. This process applies to all absolute value equations and helps us define the solution set.

Once we have found the solutions, it is crucial to verify their accuracy by substituting them into the original equation. This step allows us to check if the solutions are true for the equation. If the substituted values satisfy the equation, then they are valid solutions.

In conclusion, solving an absolute value equation requires identifying the positive and negative cases, finding the solutions, and verifying their accuracy. By following these three essential steps, we can confidently define the solution set for any absolute value equation and accurately solve similar equations in the future.

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