Inequality Problem from Spivak's Calculus: Chapter 1, Problem 4, Subproblem XI

In summary, the conversation discusses finding all values of x for which 2x<8 is true. The solution is x=-\infty, ..., 0, \pm 1, \pm 2, \pm 3. The conversation also discusses using the fact that 8=2^{3} and the increasing nature of 2^{x} to prove the solution.
  • #1
Why?
6
0

Homework Statement


Find all numbers x for which:

2x<8

Homework Equations


The Attempt at a Solution



I really haven't been able to figure this one out.
 
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  • #2
Do you have any calculations to show?

I assume it means integer values? If so surely isn't it just:

[tex]x=-\infty, ..., 0, \pm 1, \pm 2, \pm 3[/tex]

Unless I'm missing the point somewhere? It seems a bit simple though.

:smile:
 
  • #3
Note that [tex]8=2^{3}[/tex], 2^{x} in monotonically increasing. So the question is, what values of x satisfy
[tex]
2^{x}<2^{3}
[/tex]
can you say what values satisfy this equation?
 
  • #4
Stated in a different way, [itex]\log_2(t)[/itex] is an increasing function. Inequalities remain true if you apply an increasing function.
 
  • #5
Sorry I wasn't very clear.

Just thinking it through I know that 23 is 8, so x<3.

However my difficulty was in proving it, using the mathematical context that Spivak uses.

hunt_mat's example makes a lot of sense to me and fulfills that need to explain it more concretely.

Thanks for your help!
 

1. What is the "Inequality Problem" from Spivak's Calculus?

The "Inequality Problem" from Spivak's Calculus is a problem from Chapter 1, Problem 4, Subproblem XI of the textbook "Calculus" by Michael Spivak. It involves solving a series of inequalities using basic algebra and properties of inequalities.

2. Why is this problem important in calculus?

This problem is important in calculus because it helps students develop their understanding of inequalities and how they relate to concepts in calculus, such as limits and derivatives. It also serves as a good foundation for more complex problems and topics in calculus.

3. What is Subproblem XI and how does it differ from the other subproblems?

Subproblem XI is the last subproblem in Problem 4 of Chapter 1. It differs from the other subproblems because it involves solving a system of three inequalities, while the other subproblems only involve solving one or two inequalities.

4. What are some key strategies for solving this problem?

Some key strategies for solving this problem include identifying the variables and their relationships, using algebraic properties of inequalities, and setting up a system of equations to solve for the variables. It is also important to carefully check the solution and make sure it satisfies all of the given inequalities.

5. How can I check my work for this problem?

You can check your work for this problem by plugging the solution into each of the given inequalities and making sure that it satisfies all of them. You can also graph the inequalities on a coordinate plane to visually check if the solution lies within the shaded region. Additionally, you can use an online inequality calculator or ask a teacher or tutor to review your work.

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