Hi there! I'm actually just learning about absolute value equations right now, so I'll walk you through how to solve these types of problems, because I know they can get confusing.
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7. There is an error in the student work shown below: Solve |x - 1| - 3 = 5
The student split the equation into two equations. She split it into x - 1 - 3 = 5 or x - 1 - 3 = -5. We can already see there is an error in what the student did.
Before splitting your absolute value equation into two cases, a positive case and a negative case, you must make sure you isolate your absolute value.
The student should have added 3 to both sides to isolate |x - 1|. Since she did not do that, her work will be off because of her minor error.
Now let's solve for the actual answer. You have |x - 1| - 3 = 5. Let's not make the mistake that the student made, and add 3 to both sides of the equation to isolate the absolute value. Now you should have:
|x - 1| = 8
Split this absolute value equation into two cases, a positive case and a negative case. I assume you've learned about this. You can now remove the absolute value symbols. Your cases should look like:
x - 1 = 8 OR x - 1 = -8
Add 1 to both sides in both of the equations that you have made. Now you should have two answers to the absolute value problem, that should both work.
x = 9 OR x = -7; so the answer is x = 9, -7.
<em>--- Checking work ---</em>
I always check my work because sometimes absolute value equations do not hold true. Substitute 9 into the isolated absolute value equation.
|x - 1| = 8 becomes |(9) - 1| = 8
|(9) - 1| = 8 is true, because you solve inside the absolute value equation and you get 8 = 8, which is a true statement. Now let's substitute -7 into the isolated absolute value equation.
|x - 1| = 8 becomes |(-7) - 1| = 8
|(-7) - 1| = 8 is true, because you solve inside the absolute value equation and you get |-8| = 8, which is the same as 8 = 8, so it is a true statement. Both of the answers we got are valid answers.
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8. |6 - 4r| + 5 = 0
You know what to do first always, isolate the absolute value. Subtract 5 from both sides of the equation. Now you should have:
|6 - 4r| = -5
Now you can split this absolute value equation into two cases, which will be a positive case and a negative case. You can also remove the absolute value symbols. Now you should have:
6 - 4r = -5 OR 6 - 4r = 5
Subtract 6 from both sides of the equation in both of the equations. Rewrite your equation, and now you should have:
-4r = -11 OR -4r = -1
Divide both sides of the equation by -4 in both of the equations. After performing this operation, you should have both of your answers.
r = 11/4 OR r = 1/4
Substituting both answers will result in false statements, so that means your answer is no solution.
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9. |8 + Y| = 2Y - 3
Since the absolute value is already isolated, you know the drill, let's split the equation into two cases, one positive case and one negative case. Now you should have these two equations: (remember to add parentheses around the negative case since both numbers need to get multiplied by -1)
8 + Y = 2Y - 3 OR 8 + Y = -(2Y - 3)
Let's solve the positive case first (first case). Let's start by subtracting 8 from both sides of the equation. Rewrite and now you should have:
Y = 2Y - 11
Now let's subtract 2Y from both sides of the equation. Rewrite and now you should have:
-Y = -11
Divide both sides by negative one to make Y positive. Your final answer should be:
Y = 11
Now let's solve the negative case (second case). Let's start by distributing the negative sign (-1) inside the parentheses. Now you should have:
8 + Y = -2Y + 3
Subtract 8 from both sides of the equation. Rewrite and you should have:
Y = -2Y - 5
Add 2Y to both sides of the equation. Rewrite and you should have:
3Y = -5
To finish off this problem and solve for/isolate Y, divide both sides by 3. Rewrite the equation and your final answer should look like:
Y = -5/3
Substituting -5/3 and 11 will result in a false statement for -5/3 and a true statement for 11, so the answer is Y = 11.
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<em>--- Final answers ---</em>
7. The error is that the student did not isolate the absolute value first, and she should have added 3 to both sides of the equation first. The answer is x = 9, -7.
8. The answer is no solution.
9. The answer is Y = 11.
10. The answer is v = -8/3.
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*I had to cut question 10 out because I exceeded the limit of characters, sorry! Just for future reference, the limit is 3 problems per question.*
Hope this helped you! Please feel free to comment below if you have any questions (or pm me) and I'll try to get back to you! Good luck on your homework.
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