The number is closer to 60 than to 70. It is one number closer to65 than 68 is.
63
To find the number that fulfills these conditions, let's break down the given information step by step.
1. "The number is closer to 60 than to 70."
To determine whether a number is closer to 60 or 70, we can calculate the absolute difference between the number and each of these values. The number will be closer to 60 if the absolute difference between the number and 60 is smaller than the absolute difference between the number and 70.
Let's represent the number as 'x'. Therefore, the two absolute differences will be:
Absolute difference between x and 60: |x - 60|
Absolute difference between x and 70: |x - 70|
In this case, since the number is closer to 60, we can write the inequality:
|x - 60| < |x - 70|
2. "It is one number closer to 65 than 68 is."
Similar to step 1, we need to calculate the two absolute differences: one between x and 65, and the other between 68 and 65. The number will be one unit closer to 65 than 68 if the absolute difference between x and 65 is one unit smaller than the absolute difference between 68 and 65.
Absolute difference between x and 65: |x - 65|
Absolute difference between 68 and 65: |68 - 65|
In this case, since the number is one unit closer to 65, we can write the inequality:
|x - 65| < |68 - 65|
Now we have two inequalities:
1. |x - 60| < |x - 70|
2. |x - 65| < |68 - 65|
To solve these inequalities simultaneously, we can plot the number line and identify the values that satisfy both conditions. However, if you're looking for the specific number that satisfies both inequalities, we can solve them algebraically.
1. Solving the first inequality:
- Case 1: (x - 60) < (x - 70)
Solving this inequality, we find that there is no specific solution. It implies x < x, which is not possible. Thus, we move to case 2.
- Case 2: -(x - 60) < x - 70
Simplifying, we get -x + 60 < x - 70
Rearranging, 130 < 2x
Dividing by 2, we find x > 65
2. Solving the second inequality:
- Case 3: (x - 65) < (68 - 65)
Simplifying, we get x - 65 < 3
Adding 65 to both sides, we find x < 68
Therefore, the number 'x' that satisfies both conditions is greater than 65 and less than 68.