Match the inequalities with the corresponding statements.

[name_ask17]

Homework Statement

PROBLEM: Match the inequalities with the corresponding statements.
INEQUALITIES: 1) |a-5|< 1/3
2) |a- 1/3|< 5
STATEMENTS: a) The distance from a to 5 is less than 1/3
b) a is less than 5 units from 1/3

The Attempt at a Solution

I solved for a for both problems and for both problems I am getting 14/3 < a< 16/3.
My problem is that since I am getting the same solution for both problems, then aren't both the statements (a and b) correct for both of the inequalities? Or am I getting the wrong answers when I solve the inequalities? Please advise if you see the problem. Thanks in advance.

 

[vela]

It seems you've solved the inequalities incorrectly, but solving them really isn't necessary. You want to be able to interpret the inequalities as written. For example, what does |a-5| correspond to geometrically?

 

[SammyS]

Of course, vela is correct, but if you want to check what's wrong with your algebraic method, show how you get the answer for each inequality.

 

[name_ask17]

VELA: Geometrically, |a-5| corresponds to moving a over right 5, correct? But how does that help me interpret the inequality as written? Or am I interpreting it incorrecly. Please advise.

SAMMY: This is how I worked the first one out algebraically.
|a-5|< 1/3 has two solutions right? One positive and one negative?
Solution 1:
a-5 < 1/3
a< 1/3 + 5
a < 16/3

and a-5 > -1/3
a > -1/3 + 5
a > 14/3

So that gives me 14/3 < a< 16/3.
And I worked out the second problem with the same steps to get the same answer. I'm confused.

 

[ehild]

So that gives me 14/3 < a< 16/3.
And I worked out the second problem with the same steps to get the same answer. I'm confused.

You made a mistake:
|a-1/3|<5 means -5<a-1/3<5. Adding 5 to all sides, you get a negative number on the left. You just missed that minus sign.

ehild

 

[Mark44]

VELA: Geometrically, |a-5| corresponds to moving a over right 5, correct? But how does that help me interpret the inequality as written? Or am I interpreting it incorrecly. Please advise.

No, |a - 5| represents the distance between a and 5. Looking at things in terms of transformations, which you seem to be doing, the graph of y = |x - 5| can be seen as the translation of the graph of y = |x| by 5 units to the right.

SAMMY: This is how I worked the first one out algebraically.
|a-5|< 1/3 has two solutions right? One positive and one negative?
Solution 1:
a-5 < 1/3
a< 1/3 + 5
a < 16/3

and a-5 > -1/3
a > -1/3 + 5
a > 14/3

So that gives me 14/3 < a< 16/3.
And I worked out the second problem with the same steps to get the same answer. I'm confused.

 

[vela]

VELA: Geometrically, |a-5| corresponds to moving a over right 5, correct? But how does that help me interpret the inequality as written? Or am I interpreting it incorrecly. Please advise.

No. |a-5| is equal to the distance between a and 5. For example, when a=4 which is a distance of 1 away from 5 on the number line, you get |a-5|=1. Similarly, a=6.5, which is 1.5 more than 5, you get |a-5|=1.5.

So if you had an inequality like |x-1| < 2, you can interpret that as

|x-1|  =  "the distance between [i]x[/i] and 1"
  <    =  "is less than"
  2    =  "two"

 

[name_ask17]

ahh. just what i needed. thanks to all(: