Question

Transform the absolute value inequality into a compound inequality |6x+7| is less than or equal to 5. can someone plz help me?

Answer 1

\(\bf |6x+7| \le 5\implies
\begin{cases}
+(6x+7) \le 5\\ \quad \\
\bf -(6x+7) \le 5
\end{cases}\implies -5\le 6x+7 \le 5\)

Answer 2

each absolute value expression has 2 scenarios, thus you'd end up with 2 values for "x"
one for the positive scenario, and one for the negative scenario

Answer 3

what is a compound inequality?

Answer 4

compound = composed of more than 1 element

Answer 5

so my answer would just be 6x+7 is less than or equal to 5
and 6x+7 is greater than or equal to -5 ?

Answer 6

yeap

Answer 7

but thats just rewriting the equation into two mini equations.

Answer 8

yes it's

Answer 9

it's all you're asked to do though

Answer 10

I mean, you can solve for "x" if you want. you'd get 2 values 1 for each scenario, but you're not asked to do so this time

Answer 11

so then why did you write -5<6x+7<5? instead of two mini equations?

Answer 12

with the equal signs included, sorry i didnt know how to write them

Answer 13

hmm I did write the 2 "mini equations" just in a bit different form \(|6x+7| \le 5\implies
\begin{cases}
\color{blue}{+(6x+7) \le 5}\\ \quad \\
\color{blue}{-(6x+7) \le 5}
\end{cases}\implies -5\le 6x+7 \le 5\)

Answer 14

but why are the positive and negative signs not in front of the 5?

Answer 15

and are we supposed to flip the equation for -5?

Answer 16

yeap, because you're multiplying for a negative number, so \(\bf |6x+7| \le 5\\\implies
\begin{cases}
+(6x+7) \le 5\implies 6x+7\le 5\\ \quad \\
\bf -(6x+7) \le 5\implies -1(-(6x+7)) \le -1(5)\implies 6x+7\ge -5
\end{cases}\)

Answer 17

shoot got ..truncated... \(\bf |6x+7| \le 5\\\quad \\
\begin{cases}
+(6x+7) \le 5\implies 6x+7\le 5\\ \quad \\
\bf -(6x+7) \le 5\implies -1(-(6x+7)) \le -1(5)\implies 6x+7\ge -5
\end{cases}\)

Answer 18

so I can just write 6x+7 < or euqal to 5 OR 6x+7> or equal to -5

or do I have to write it like: {x| 6x+7<5 or 6x+7>_5}

Answer 19

either way looks fine to me, is just a matter of notation

Answer 20

ohhh ok thanks so much!

Answer 21

yw