Solve Absolute Value Equations: Extraneous Solutions
Hey guys! Today, we're diving into the fascinating world of solving absolute value equations, and we're going to tackle the trickiest part: extraneous solutions. Extraneous solutions are like those uninvited guests at a party—they seem to fit in at first, but they don't really belong. In mathematical terms, these are solutions that we find through our solving process, but they don't actually satisfy the original equation. So, it's super important to check every solution we get to make sure it's a real deal and not an imposter. We'll walk through several examples step by step, so you'll get the hang of it in no time. Let's get started!
1. Solving the Absolute Value Equation: |4x + 6| = 26
Let's kick things off with our first equation: |4x + 6| = 26. Absolute value equations can seem a bit intimidating, but don't worry, they're totally manageable once you understand the basic principle. The absolute value of a number is its distance from zero, which means it can be either positive or negative. So, to solve an absolute value equation, we need to consider both possibilities.
First, consider the expression inside the absolute value, 4x + 6, can be equal to 26. This gives us the equation:
4x + 6 = 26
To solve for x, we'll subtract 6 from both sides:
4x = 26 - 6 4x = 20
Now, divide both sides by 4:
x = 20 / 4 x = 5
So, one potential solution is x = 5. But hold on, we're not done yet! We also need to consider the possibility that the expression inside the absolute value, 4x + 6, could be equal to -26, since the absolute value of -26 is also 26. This gives us the second equation:
4x + 6 = -26
Again, we'll subtract 6 from both sides:
4x = -26 - 6 4x = -32
Now, divide both sides by 4:
x = -32 / 4 x = -8
So, our second potential solution is x = -8. Now comes the crucial step: checking for extraneous solutions. We need to plug both values back into the original equation to see if they actually work.
Let's check x = 5:
|4(5) + 6| = |20 + 6| = |26| = 26
Great! x = 5 checks out.
Now, let's check x = -8:
|4(-8) + 6| = |-32 + 6| = |-26| = 26
Awesome! x = -8 also works. So, in this case, we have two valid solutions: x = 5 and x = -8. No extraneous solutions here!
2. Solving the Absolute Value Equation: |-8 - 5r| / 6 = 2
Next up, we have the equation |-8 - 5r| / 6 = 2. This one looks a little more complex, but don't sweat it! We'll break it down step by step, just like before. The first thing we want to do is get rid of that fraction. To do that, we'll multiply both sides of the equation by 6:
|-8 - 5r| / 6 * 6 = 2 * 6 |-8 - 5r| = 12
Now we have a simpler absolute value equation to deal with. Remember, the expression inside the absolute value, -8 - 5r, can be equal to either 12 or -12. So, let's consider both possibilities.
First, let's set -8 - 5r equal to 12:
-8 - 5r = 12
To solve for r, we'll add 8 to both sides:
-5r = 12 + 8 -5r = 20
Now, divide both sides by -5:
r = 20 / -5 r = -4
So, one potential solution is r = -4. Now, let's consider the second possibility, where -8 - 5r is equal to -12:
-8 - 5r = -12
Again, we'll add 8 to both sides:
-5r = -12 + 8 -5r = -4
Now, divide both sides by -5:
r = -4 / -5 r = 4/5
So, our second potential solution is r = 4/5. Time to check for extraneous solutions! We'll plug both values back into the original equation.
Let's check r = -4:
|-8 - 5(-4)| / 6 = |-8 + 20| / 6 = |12| / 6 = 12 / 6 = 2
Perfect! r = -4 works.
Now, let's check r = 4/5:
|-8 - 5(4/5)| / 6 = |-8 - 4| / 6 = |-12| / 6 = 12 / 6 = 2
Fantastic! r = 4/5 also checks out. So, we have two valid solutions: r = -4 and r = 4/5. No extraneous solutions here either!
3. Solving the Absolute Value Equation: 8 - |7y - 9| = 3
Alright, let's move on to our third equation: 8 - |7y - 9| = 3. This one has a slight twist because the absolute value expression is being subtracted. No problem, we can handle it! Our first step is to isolate the absolute value term. To do that, we'll subtract 8 from both sides of the equation:
8 - |7y - 9| - 8 = 3 - 8 -|7y - 9| = -5
Now, we want to get rid of that negative sign in front of the absolute value. We can do this by multiplying both sides of the equation by -1:
(-1) * -|7y - 9| = (-1) * -5 |7y - 9| = 5
Great! Now we have a standard absolute value equation. The expression inside the absolute value, 7y - 9, can be equal to either 5 or -5. Let's consider both scenarios.
First, let's set 7y - 9 equal to 5:
7y - 9 = 5
To solve for y, we'll add 9 to both sides:
7y = 5 + 9 7y = 14
Now, divide both sides by 7:
y = 14 / 7 y = 2
So, one potential solution is y = 2. Now, let's consider the second possibility, where 7y - 9 is equal to -5:
7y - 9 = -5
Again, we'll add 9 to both sides:
7y = -5 + 9 7y = 4
Now, divide both sides by 7:
y = 4 / 7
So, our second potential solution is y = 4/7. Let's check for extraneous solutions by plugging both values back into the original equation.
Let's check y = 2:
8 - |7(2) - 9| = 8 - |14 - 9| = 8 - |5| = 8 - 5 = 3
Awesome! y = 2 works perfectly.
Now, let's check y = 4/7:
8 - |7(4/7) - 9| = 8 - |4 - 9| = 8 - |-5| = 8 - 5 = 3
Fantastic! y = 4/7 also checks out. So, we have two valid solutions: y = 2 and y = 4/7. No extraneous solutions here either!
4. Solving the Absolute Value Equation: |8k - 5| = 4k - 23
Last but not least, we have the equation |8k - 5| = 4k - 23. This one is particularly interesting because the expression on the right side, 4k - 23, involves a variable. This means we need to be extra careful about extraneous solutions. Let's dive in!
As usual, the expression inside the absolute value, 8k - 5, can be equal to either 4k - 23 or the negation of 4k - 23. Let's consider both possibilities.
First, let's set 8k - 5 equal to 4k - 23:
8k - 5 = 4k - 23
To solve for k, we'll subtract 4k from both sides:
8k - 4k - 5 = -23 4k - 5 = -23
Now, add 5 to both sides:
4k = -23 + 5 4k = -18
Finally, divide both sides by 4:
k = -18 / 4 k = -9/2
So, one potential solution is k = -9/2. Now, let's consider the second possibility, where 8k - 5 is equal to the negation of 4k - 23. The negation of 4k - 23 is -(4k - 23), which simplifies to -4k + 23. So, we have:
8k - 5 = -4k + 23
To solve for k, we'll add 4k to both sides:
8k + 4k - 5 = 23 12k - 5 = 23
Now, add 5 to both sides:
12k = 23 + 5 12k = 28
Finally, divide both sides by 12:
k = 28 / 12 k = 7/3
So, our second potential solution is k = 7/3. Now comes the critical part: checking for extraneous solutions. We need to plug both values back into the original equation and see if they hold true.
Let's check k = -9/2:
|8(-9/2) - 5| = 4(-9/2) - 23 |-36 - 5| = -18 - 23 |-41| = -41 41 = -41
Oops! This is not true. So, k = -9/2 is an extraneous solution. It doesn't actually satisfy the original equation.
Now, let's check k = 7/3:
|8(7/3) - 5| = 4(7/3) - 23 |56/3 - 15/3| = 28/3 - 69/3 |41/3| = -41/3 41/3 = -41/3
Again, this is not true! So, k = 7/3 is also an extraneous solution. In this case, we have no valid solutions. Both potential solutions turned out to be imposters.
And there you have it, folks! We've worked through several examples of solving absolute value equations, and we've learned how to spot and deal with those pesky extraneous solutions. Remember, the key is to always check your answers by plugging them back into the original equation. It might seem like an extra step, but it can save you from making mistakes and ensure that you find the correct solutions. Keep practicing, and you'll become a pro at solving absolute value equations in no time!
