Absolute Values (part 1)
We’ll cover the four steps to success for solving problems like this!
What does |-7| equal? That is right, 7. If you need a refresher on this, please check out my quick refresher of absolute values.
Now, solve for x:
2 + x = 3 (That’s right, there is no absolute value here, this is a warm-up!)
2 + x = 3
-2 -2
x = 1
Great! Now solve for x:
|2 + x| = 3
Let’s work through this one together.
Step 1: Write the equation twice without the absolute value bars. (Trust me!)
2 + x = 3
2 + x = 3
Step 2: With one of the equations, for everything that was inside the absolute value bars, change the sign to the opposite.
-2 - x = 3
2 + x = 3
Caution: it is very easy to make a sign mistake. Make sure you’ve changed the sign for every term that had been inside the absolute value bars. So the 2 and the x, which were both positive, go negative.
Step 3: Solve each equation for the unknown (in this case, x).
First I’ll solve -2 - x = 3
-2 - x = 3
+2 +2
-x = 5
x = -5
Now I’ll solve this one: 2 + x = 3
2 + x = 3
-2 -2
x = 1
So now we know x = -5 and x = 1.
Step 4: Check your work! Substitute the answers you got back into the original equation. It feels so good to know you are right. How often does life give you that chance?
For |2 + x| = 3, we state our answers are x = -5 and x = 1
First I’ll substitute in -5 for x
|2 + x| = 3
|2 + (-5)| = 3
|-3| = 3
That is correct! |-3| does equal 3.
Now I’ll substitute in 1 for x
|2 + x| = 3
|2 + 1| = 3
|3| = 3
That is correct! Wow you are good.
Try this one, using the steps for success!
|2y -3| = 5
Try it on your own, then check your work with the solution below.
Steps to Success
Solving absolute value problems
Step 1: Write the equation twice without the absolute value bars.
Step 2: With one of the equations, for everything that was inside the absolute value bars, change the sign to the opposite.
Step 3: Solve each equation for the unknown.
Step 4: Check your work! Substitute the answers you got back into the original equation.
|2y -3| = 5
Step 1: Write the equation twice without the absolute value bars.
2y – 3 = 5
2y – 3 = 5
Step 2: With one of the equations, for everything that was inside the absolute value bars, change the sign to the opposite.
-2y + 3 = 5
2y – 3 = 5
For the top equation, we changed the sign for every term that had been inside the absolute value bars. The 2y, which was positive, became negative. The 3, which was negative, became positive.
Step 3: Solve each equation for the unknown (in this case, y).
First I’ll solve -2y + 3 = 5
-2y + 3 = 5
-3 -3
-2y = 2
y = -1
Now I’ll solve 2y – 3 = 5
2y – 3 = 5
+3 +3
2y = 8
y = 4
So now we know y = -1 and y = 4.
Step 4: Check your work! Substitute the answers you got back into the original equation.
For |2y -3| = 5, we state our answers are y = -1 and y = 4.
First I’ll substitute in -1 for y
|2y -3| = 5
|2(-1) -3| = 5
|-2 -3| = 5
|-5| = 5
Correct! You are on a roll.
Now I’ll substitute in 4 for y
|2y -3| = 5
|2(4) -3| = 5
|8 -3| = 5
|5| = 5
Fantastic. Your answers are correct. y = -1 and y = 4. Congrats!