Solve f(1/2−3|x|)+f(5)>0.

01 / Standard#Your turn

f(t)=\begin{cases}\log_2t&0<t\le4\\t^2-8t+17&t>4\end{cases}

Official paper · jm01-2023 · I.8 · PDF 2

Skills and prerequisite lessons

Option A $(-1/12,1/12)$
Option B $(-1/6,1/6)$
Option C $(-1/4,1/4)$
Option D $(-1/3,1/3)$
Option E $(-1/2,1/2)$

Working and explanation

BUILD THE REASONING

Hint 1

Compute f(5) from the second branch.

Hint 2

The other input is at most 1/2 and must be positive.

Worked solution

Use the logarithmic branch with its domain.
$f(5)=2,\quad\log_2(1/2-3|x|)>-2$
Since the base is greater than one, preserve the inequality direction.
$1/2-3|x|>1/4\iff|x|<1/12$

A: −1/12<x<1/12.

Checks and common pitfalls: The positive logarithm argument is automatically enforced by the stronger final bound.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Use the logarithmic branch with its domain.
$f(5)=2,\quad\log_2(1/2-3|x|)>-2$
Since the base is greater than one, preserve the inequality direction.
$1/2-3|x|>1/4\iff|x|<1/12$

Think first. Reveal a hint when the class is ready.

2023 JM01