Hello, Henryt999!

For what positive number $\displaystyle s$ is the following statements true?

. . $\displaystyle \begin{array}{cccc}|x-3| &<& s & [1]\\ \\[-3mm] |x-4| &<& 2 & [2] \end{array}$ Note that: .$\displaystyle s \,\geq\,0$

From [1], we have: .$\displaystyle -s \:<\:x-3 \:<\:s \quad\Rightarrow\quad -s+3 \:<\:x\:<\:s + 3 $ .[3]

From [2], we have: .$\displaystyle -2 \:<\:x-4\:<\:2 \quad\Rightarrow\quad 2 \:<\:x\:<\:6 $ .[4]

[3] tells us that $\displaystyle x$ is between $\displaystyle 3-s$ and $\displaystyle 3+s$ Code:

- - - o = = = = = = o - - -
3-s 3+s

[4] tells us that $\displaystyle x$ is between 2 and 6. Code:

- - o = = = = = = = = o - -
2 6

For both statements to be true,

. . the first interval must be "inside" the second interval. Code:

- - - o = = = = = = o - - -
3-s 3+s
- - o = = = = = = = = o - -
2 6

So we have: .$\displaystyle \begin{array}{ccccccc}3-s \:\geq\:2 & \Rightarrow & -s \:\geq \:-1 & \Rightarrow & s \:\leq\:1 \\

3+s \:\leq\:6 & \Rightarrow & s \:\leq\:3 \end{array}$

. . which are both true for: .$\displaystyle s \,\leq\,1$

Therefore, the inequalities are true for: .$\displaystyle 0 \,\leq s\,\leq\,1$