For f(x)=|3x-17|, what is one possible value of a for which f(a)<a?

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- May 3rd 2012, 08:50 PMmathDadAbsolute value function (SAT Math Question)
For f(x)=|3x-17|, what is one possible value of a for which f(a)<a?

- May 4th 2012, 12:31 AMProve ItRe: Absolute value function (SAT Math Question)
Can you solve $\displaystyle \displaystyle \begin{align*} |3a-17| &< a \end{align*}$?

- May 4th 2012, 04:43 AMemakarovRe: Absolute value function (SAT Math Question)
Or look at the graph.

Attachment 23780 - May 4th 2012, 08:46 AMmathDadRe: Absolute value function (SAT Math Question)
Reply to ProveIt:

We tried that and couldn't figure it out. I think this is what we did:

$\displaystyle |3a-17|<a$

$\displaystyle -a<3a-17<a$

$\displaystyle -a+17<3a<a+17$

$\displaystyle \frac{-3+17}3<a<\frac{a+17}3$

Not sure what to do from here - May 4th 2012, 08:48 AMmathDadRe: Absolute value function (SAT Math Question)
In reply to emakarov:

That narrows it down, but not sure if you can get a good enough answer from that on the test (if she can use a calc on it) - May 4th 2012, 08:54 AMemakarovRe: Absolute value function (SAT Math Question)
By definition, -a < 3a - 17 < a is two inequalities: -a < 3a - 17 and 3a - 17 < a. Solve both inequalities, take the intersection of the sets of their solutions (i.e., find the set of a's that are solutions to both inequalities) and pick any element of the intersection (since the question asks for "one possible value").

- May 4th 2012, 08:59 AMemakarovRe: Absolute value function (SAT Math Question)
The graph can give you an idea for which x to try. Then you can pick an x and simply by calculation prove that it satisfies the inequality.

In the proofs of existence, you have to provide a witness (x in this case) and a proof that the witness satisfies the necessary property. You don't have to explain how you picked the witness unless it is a requirement of this particular problem/instructor/course. - May 4th 2012, 09:03 AMPlatoRe: Absolute value function (SAT Math Question)
$\displaystyle |3a-17|<a$

$\displaystyle -a<3a-17<a$ At this point split the expression into two parts.

$\displaystyle -a<3a-17~~\&~~3a-17<a~~$ solve each and intersect the solutions. - May 4th 2012, 09:07 AMmathDadRe: Absolute value function (SAT Math Question)
- May 4th 2012, 09:08 AMmathDadRe: Absolute value function (SAT Math Question)