Think of -1.1 as -11
Think of -1.2 as -12
I moved the decimal points to the right for each value
Then on a vertical number line, notice how -12 is below -11, so -12 < -11
If we divide both sides by 10, then we go from -12 < -11 to -1.2 < -1.1
In short, the final expression you have -1.1 < middleStuff < -1.2 is in the wrong order.
It should be -1.2 < 2 - sqrt(10) < -1.1
I swapped the places of -1.1 and -1.2
If you were to use a calculator, then 2-sqrt(10) = -1.162 approximately which fits the description of being between -1.2 and -1.1; this helps confirm we have the correct upper lower and upper bounds.
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Here's another way to look at it
3.1 < sqrt(10) < 3.2
-1*3.1 > -1*sqrt(10) > -1*3.2 ............ see note1 below
-3.1 > -sqrt(10) > -3.2
-3.1+2 > -sqrt(10)+2 > -3.2+2 ....... see note2
2-3.1 > 2-sqrt(10) > 2-3.2
-1.1 > 2 - sqrt(10) > -1.2
-1.2 < 2 - sqrt(10) < -1.1 ........... see note3
Footnotes
- note1: For this step I multiplied all three sides by -1. Multiplying by a negative will flip the inequality signs
- note2: I added 2 to all three sides. This does not flip any of the inequality signs.
- note3: We go from the form a > b > c to c < b < a. The outer sides 'a' and 'c' swap places while b in the middle stays where it is. The inequality signs flip. It might help to think of something like 3 > 2 > 1 flipping to 1 < 2 < 3.