MHB How Do Absolute Values Express At Least and At Most Conditions?

AI Thread Summary
Absolute values are used to express conditions of "at least" and "at most" by defining the distance between two values. For "at least" statements, such as the distance between x and 4 being at least 8, the correct expression is |x - 4| ≥ 8, indicating that the value is equal to or greater than 8. Conversely, for "at most" statements, like the distance between x^3 and -1 being at most 0.001, the expression is |x^3 - (-1)| ≤ 0.001, meaning the value is equal to or less than 0.001. Understanding these distinctions is crucial for accurately applying inequalities in mathematical contexts. This knowledge is essential for solving problems involving absolute values effectively.
mathdad
Messages
1,280
Reaction score
0
Rewrite each statement using absolute values.

1. The distance between x and 4 is at least 8.

Work:

| x - 4 | > or = 8

Correct?

Why must we write greater than or equal to for AT LEAST statements?

2. The distance between x^3 and -1 is at most 0.001.

Work:

| x^3 - (-1) | < or = 0.001

Correct?

Why must we write less than or equal to for AT MOST statements?
 
Mathematics news on Phys.org
RTCNTC said:
Rewrite each statement using absolute values.

1. The distance between x and 4 is at least 8.

Work:

| x - 4 | > or = 8

Correct?

Why must we write greater than or equal to for AT LEAST statements?

Yes, that's correct. When we say something is "at least" some value, that's equivalent to saying it is that value or greater. If I say I have at least \$20 in my pocket, then you know the money in my pocket is \$20 or more.

RTCNTC said:
2. The distance between x^3 and -1 is at most 0.001.

Work:

| x^3 - (-1) | < or = 0.001

Correct?

Why must we write less than or equal to for AT MOST statements?

That's correct too. When we say some value is at most some other value, then that's equivalent to saying it is that value or less. If I say I have "at most" \$20 in my pocket then you know the money I have in my pocket is less than or equal to \$20. :D
 
Good to know the difference between "at least" and "at most" because it is very common in the world of inequality applications.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top