How do you solve and write the following in interval notation: # | (3x) / 4 + 5 | < 1#?
x < -5
When solving inequalities inside of absolute value bars, you simply get rid of the bars the first time:
The first step would be subtracting 5 from both sides because the 5 is positive. Doing this cancels the 5 out on both sides and gives you a -4 on the right side of the inequality. 3x/4 < -4
Next, you would take -4 times 4 to cancel out the four on the left side of the inequality. This gives you 3x < -16
After doing this, you simply divide the -16 by 3. Doing this, you can get one of two answers: decimal or fraction. The decimal is repeating giving you -5.333333333....., which is why it would be a better option to use the value -5
For the second answer, you take all of the values inside of the absolute value bars and change their signs. 3x would become -3x, 4 would become -4, and 5 would become -5. (-3x/4-5 < 1)
You would then remove the absolute value bars and solve normally.
First, you add 5 to each side to cancel 5 out on the left. This gives you -3x/-4 <6.
Next, you would multiply -4 by 6 and get -24. The problem is now -3x < -24.
When solving inequalities, there is a special rule when there is a negative number with a variable. Whenever there is a negative number with a variable, you solve normally, but you flip your sign in the middle. In this case, the < would become a > sign.
You should take -24 divided by -3 and the answer to the second problem should be x > 8.
I hope this was helpful :)