Solve each equation. Check your answers. |4-z|-10=1 |...

Name: Problem 9, Chapter 1: Algebra 2
Uploaded: 2020-02-22T04:40:44-08:00
Duration: 4 min 50 s
Channel: Kelsey Van Horn
Description: Solve each equation. Check your answers. |4-z|-10=1

Key Concepts

Absolute Value Function

The absolute value function measures the distance of a number from zero on the number line, regardless of direction. It is defined as |x|, which is equal to x if x is non-negative and -x if x is negative. This concept is critical when dealing with equations that include absolute values, as it underpins the process of removing the absolute value sign by considering both its positive and negative scenarios.

Solving Absolute Value Equations with Multiple Cases

When solving an equation that involves an absolute value, the usual strategy is to isolate the absolute value expression and then create two separate cases: one where the expression inside the absolute value is equal to the positive value, and one where it is equal to the negative value. This two-case approach is essential because the absolute value of an expression is always non-negative, and so the equation can be split accordingly to capture all possible solutions.

Checking for Extraneous Solutions

After finding potential solutions by solving the cases of an absolute value equation, it is important to substitute these solutions back into the original equation to verify their validity. This process of checking is crucial because the act of squaring both sides or splitting into cases can sometimes introduce solutions that do not actually satisfy the original equation.

Transcript

00:01 Okay, so for this problem, we are asked to solve the equation, the absolute value of 4 minus z minus 10 equals 1.

00:09 So first we can sort of treat this absolute value portion like parentheses, and we want to get this by itself.

00:16 So we'll start off by adding 10 to each side.

00:20 And so we end up with the absolute value of 4 minus z equals 11.

00:26 Now the absolute value is just the distance a number is from zero.

00:31 So it can be negative or positive inside the absolute value symbol, but the absolute value of it will always be positive.

00:40 So for this example, the absolute value of 11 is 11 because it's 11 from 0.

00:46 So that would give us 11 equal to 11.

00:49 So we know 11, the quantity of 4 minus z has to equal 11.

00:53 And negative 11 can also be a potential answer because negative 11 is also 11 from 0.

01:00 So 4 minus z can also equal negative 11.

01:05 So once we do once we figure out what our two possible answers are for what lies inside, the absolute value symbols, negative 11 and 11, we'll set the equation inside of the absolute value symbol to equal both negative 11 and positive 11.

01:29 And from here, we can just do our simple algebra, subtract four from each side to get z alone, and we end up with negative z equal to negative 11 minus 4, which is negative 15.

01:47 And we can divide each side by negative 1 to get rid of this negative sign on z, and we end up with z equal to 15.

01:58 And then if we finish solving over here, these will cancel out, and we get negative z equals 11 minus 4, which is 7...

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