Answer :
To solve this problem, we want to determine the range of acceptable weights [tex]\(x\)[/tex] for the suitcase, knowing that it can vary by at most 7.5 pounds from the desired weight of 40 pounds.
### Step-by-Step Solution
1. Define the Acceptable Deviation:
- The weight of the suitcase can vary by at most 7.5 pounds from 40 pounds.
2. Set Up the Inequality:
- We express this as an absolute value inequality. If [tex]\(x\)[/tex] is the weight of the suitcase, then the deviation from 40 pounds should be within 7.5 pounds. Mathematically, this is represented as:
[tex]\[
|x - 40| \leq 7.5
\][/tex]
3. Solve the Absolute Value Inequality:
- An absolute value inequality of the form [tex]\(|A| \leq B\)[/tex] translates to two separate inequalities:
[tex]\[
-B \leq A \leq B
\][/tex]
- Applying this to our inequality:
[tex]\[
-7.5 \leq x - 40 \leq 7.5
\][/tex]
4. Isolate [tex]\(x\)[/tex]:
- To find the range for [tex]\(x\)[/tex], we solve these two inequalities separately.
For the lower bound:
[tex]\[
-7.5 \leq x - 40
\][/tex]
Adding 40 to both sides:
[tex]\[
32.5 \leq x
\][/tex]
For the upper bound:
[tex]\[
x - 40 \leq 7.5
\][/tex]
Adding 40 to both sides:
[tex]\[
x \leq 47.5
\][/tex]
5. Combine the Results:
- Combining both results, we get the range for [tex]\(x\)[/tex]:
[tex]\[
32.5 \leq x \leq 47.5
\][/tex]
### Final Answer
To find the range of acceptable weights for your suitcase where [tex]\(x\)[/tex] is the weight of the suitcase, the inequality can be written as:
[tex]\[
|x - 40| \leq 7.5
\][/tex]
And the range of acceptable weights [tex]\(x\)[/tex] is:
[tex]\[
32.5 \leq x \leq 47.5
\][/tex]
This means the suitcase can weigh anywhere between 32.5 pounds and 47.5 pounds inclusively.
### Step-by-Step Solution
1. Define the Acceptable Deviation:
- The weight of the suitcase can vary by at most 7.5 pounds from 40 pounds.
2. Set Up the Inequality:
- We express this as an absolute value inequality. If [tex]\(x\)[/tex] is the weight of the suitcase, then the deviation from 40 pounds should be within 7.5 pounds. Mathematically, this is represented as:
[tex]\[
|x - 40| \leq 7.5
\][/tex]
3. Solve the Absolute Value Inequality:
- An absolute value inequality of the form [tex]\(|A| \leq B\)[/tex] translates to two separate inequalities:
[tex]\[
-B \leq A \leq B
\][/tex]
- Applying this to our inequality:
[tex]\[
-7.5 \leq x - 40 \leq 7.5
\][/tex]
4. Isolate [tex]\(x\)[/tex]:
- To find the range for [tex]\(x\)[/tex], we solve these two inequalities separately.
For the lower bound:
[tex]\[
-7.5 \leq x - 40
\][/tex]
Adding 40 to both sides:
[tex]\[
32.5 \leq x
\][/tex]
For the upper bound:
[tex]\[
x - 40 \leq 7.5
\][/tex]
Adding 40 to both sides:
[tex]\[
x \leq 47.5
\][/tex]
5. Combine the Results:
- Combining both results, we get the range for [tex]\(x\)[/tex]:
[tex]\[
32.5 \leq x \leq 47.5
\][/tex]
### Final Answer
To find the range of acceptable weights for your suitcase where [tex]\(x\)[/tex] is the weight of the suitcase, the inequality can be written as:
[tex]\[
|x - 40| \leq 7.5
\][/tex]
And the range of acceptable weights [tex]\(x\)[/tex] is:
[tex]\[
32.5 \leq x \leq 47.5
\][/tex]
This means the suitcase can weigh anywhere between 32.5 pounds and 47.5 pounds inclusively.
