Match the inequalities with the corresponding statements.

In summary, Sammy solved the inequalities incorrectly but they are still useful for interpreting the written inequality.
  • #1
name_ask17
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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.
 
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  • #2
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?
 
  • #3
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.
 
  • #4
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.
 
  • #5
name_ask17 said:
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
 
  • #6
name_ask17 said:
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.
name_ask17 said:
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.
 
  • #7
name_ask17 said:
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

Code:
|x-1|  =  "the distance between [i]x[/i] and 1"
  <    =  "is less than"
  2    =  "two"
 
  • #8
ahh. just what i needed. thanks to all(:
 

FAQ: Match the inequalities with the corresponding statements.

1. What are inequalities?

Inequalities are mathematical expressions that compare two quantities or values. They use symbols such as <, >, ≤, and ≥ to represent that one value is greater than, less than, or equal to the other value.

2. How do you match inequalities with statements?

You can match inequalities with statements by identifying the symbols used in the inequality and understanding their meaning. For example, if the inequality symbol is <, the statement can be read as "less than".

3. Can you give an example of matching an inequality with a statement?

Sure, an example would be matching the inequality 3x + 5 < 10 with the statement "Three times a number plus five is less than ten".

4. Why is it important to understand inequalities?

Understanding inequalities is important in many areas of science and everyday life. They are used in data analysis, economics, and decision-making processes. They also help us compare and interpret data and make predictions based on patterns and relationships.

5. What are some common mistakes when matching inequalities with statements?

Some common mistakes include misinterpreting the symbols, not paying attention to the order of the values, and forgetting to include units of measurement. It's important to double check your work and make sure the statement accurately reflects the meaning of the inequality.

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