In mathematics, equations with absolute values can be intimidating at first glance. However, with the right tools and techniques, solving these types of equations can become much easier. In this article, we will explore how to solve equations with absolute values step by step.

First, it is important to understand what an absolute value is. An absolute value is a mathematical function that returns the positive value of a number, regardless of whether it is negative or positive. For example, the absolute value of -5 is 5, since the distance between -5 and 0 on a number line is 5 units.

To solve an equation with an absolute value, we need to isolate the absolute value on one side of the equation. Let’s look at an example:

|2x – 3| = 5

The first step is to separate the absolute value on one side of the equation. We can do this by writing two separate equations, one with a positive argument and one with a negative argument. This gives us:

2x – 3 = 5 or 2x – 3 = -5

Next, we solve each equation separately. Starting with the first equation, we add 3 to both sides to isolate the variable:

2x = 8

Then, we divide both sides by 2 to solve for x:

x = 4

For the second equation, we add 3 to both sides again:

2x = -2

And then divide both sides by 2:

x = -1

Therefore, the solutions to the original equation are x = 4 and x = -1.

It is important to note that when the absolute value is greater than a positive number, or less than a negative number, there are no solutions to the equation. For example:

|2x – 3| > 10

In this case, we write two separate equations as before, but with opposite inequality symbols:

2x – 3 > 10 or 2x – 3 < -10 Solving the first equation gives us: 2x > 13

x > 6.5

Solving the second equation gives us:

2x < -7 x < -3.5 Therefore, there are no solutions to the original equation, since x cannot be both greater than 6.5 and less than -3.5. When solving equations with absolute values, it is also important to consider the absolute value of a variable and not just a constant. Let's look at an example: |2x + 1| = |x - 3| In this case, we need to consider both positive and negative values for the absolute value. We write two separate equations again, but this time with different arguments: 2x + 1 = x - 3 or 2x + 1 = -(x - 3) Solving the first equation gives us: x = -4 Solving the second equation gives us: 3x = -2 x = -2/3 Therefore, the solutions to the original equation are x = -4 and x = -2/3. In summary, solving equations with absolute values may seem daunting, but with a few simple steps, it can become much easier. Always remember to isolate the absolute value, write two separate equations, solve each equation separately, and consider both positive and negative values for the absolute value. With these techniques, you'll be solving equations with absolute values like a pro in no time!

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