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Solving Absolute Value Equations

Absolute Value Practice Qs

Here's what you need to know to solve absolute value equations:

1. What is Absolute Value?

Absolute value shows how far a number is from zero on a number line. Since distance is always positive, absolute value is always a positive number. For example:

|3| = 3, because 3 is 3 units from zero.
|-3| = 3, because -3 is also 3 units from zero.

2. Steps to Solve Absolute Value Equations

When you see an equation like |x| = 3, it means two things:

x = 3 (because 3 is 3 units from zero)
x = 3 (because -3 is also 3 units from zero)
If you see something like |x| = -3, there’s no solution. Why? Because absolute value can’t be negative!

3. How to Solve Complex Absolute Value Equations

Consider |x + 2| = 5

Solve |x + 2| = 5 for the + side of the number line:

|x + 2| = 5

x + 2 = 5

-2 -2

x = 3

Solve |x + 2| = 5 for the NEGATIVE side of the number line:

|x + 2| = 5

Set 5 to negative.

x + 2 = -5

-2 -2

x = -7

4. Always Check Your Solutions

Substitute each answer back into the original equation to make sure both solutions work.

|x + 2| = 5

|3 + 2| = 5

|5| = 5

5 = 5 ✔

Substitute each answer back into the original equation to make sure both solutions work.

|x + 2| = 5

|-7 + 2| = 5

|-5| = 5

5 = 5 ✔

5. Soving More Complex Absolute Value Equations

Consider 5 + |x + 6| = 7

Solve 5 + |x + 6| = 7 for the + side of the number line:

5 + |x + 6| = 7

Subtract the 5:

5 + |x + 6| = 7

-5 -5

|x + 6| = 2

Remove the brackets and subtract the 6:

x + 6 = 2

-6 -6

x = -4

Solve 5 + |x + 6| = 7 for the NEGATIVE side of the number line:

Start with this:

|x + 6| = 2

Set the 2 to Negative:

|x + 6| = -2

Remove the brackets and subtract the 6:

x + 6 = -2

-6 -6

x = -8

Once you understand these steps, try solving different absolute value equations for practice!

Absolute Value Practice Qs

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