Find the solution set of two to the absolute value of eight 𝑥 minus 12 equals eight
to the four 𝑥 minus four.
We’ll just start by copying down our problem. When we’re solving for exponents, we have to work with exponents who have the same
base. So the first thing we do here is try to convert this eight to a power of two. I know that two to the third power equals eight. I’ll substitute two to the third power in for eight.
If we’re taking a power to a power, we need to multiply. We’ll distribute that power of three over the four 𝑥 and the negative four. Eight to the four 𝑥 minus four is the same thing as two to the 12𝑥 minus 12.
Now that we have the same bases, we can solve for 𝑥 by setting the powers equal to
each other. When working with absolute value, we work with two cases: positive eight 𝑥 minus 12
equals 12𝑥 minus 12 and negative eight 𝑥 minus 12 equals 12𝑥 minus 12.
We’ll start on the left and try to solve for 𝑥. We can take away the parentheses. And now, we have the equation eight 𝑥 minus 12 equals 12𝑥 minus 12. We can add 12 to both sides. And that leaves us with the statement eight 𝑥 equals 12𝑥. This is only true for one value: eight times zero equals 12 times zero. So there is a possible solution at 𝑥 equals zero.
We’ll move on to the other option. We need to distribute our negative value to both of our terms. Negative eight 𝑥 plus 12 equals 12𝑥 minus 12. I’m gonna take eight 𝑥 and add it to the left and to the right. Our eight 𝑥 cancels out on the left. And now, we have 12 equals 12𝑥 plus eight 𝑥 equals 20𝑥 minus 12.
We can add 12 to both sides of our equation. Negative 12 plus 12 equals zero, 12 plus 12 equals 24, 24 equals 20𝑥, and then we
divide both sides of our equation by 20. On the right, 20 divided by 20 leaves us with 𝑥. 𝑥 equals 24 over 20.
If we simplify or reduce 24 over 20, we divide 24 and 20 by four. 24 by four equals six. 20 divided by four equals five. We have two options. But we need to check and make sure they’re both valid. We do that by plugging the 𝑥-values we found back into our original statement. We’ll start by substituting zero in for 𝑥.
Here, we go. The absolute value of eight times zero minus 12 is that equal to 12 times zero minus
12. Zero minus 12 equals negative 12. 12 times zero is zero. Minus 12 equals negative 12. What’s happening here is we have the absolute value of negative 12 being equal to
negative 12. And that’s not possible. 𝑥 equals zero is not a valid option.
We can go ahead and plug in six over five into that same equation. We now have eight times six over five minus 12, the absolute value of that equal to
12 times six over five minus 12. Eight times six over five equals nine and six tenths. nine and six tenths minus 12
equals negative two and four tenths.
But we can’t forget that we’re taking the absolute value here. The absolute value of negative two and four tenths equals two and four tenths. On the other side, 12 times six over five equals 14 and four tenths minus 12 equals
two and four tenths. two and four tenths equals two and four tenths, which means
that the solution set here only includes six over five.