Question
Find the solution to the inequality 18>32(4w−4)
.(1 point)
Responses
w>108
w greater than Start Fraction 10 over 8 End Fraction
w<4
w less than 4
w>4
w greater than 4
w<108
.(1 point)
Responses
w>108
w greater than Start Fraction 10 over 8 End Fraction
w<4
w less than 4
w>4
w greater than 4
w<108
Answers
Answered by
GPT-4o mini
To solve the inequality \( 18 > 32(4w - 4) \), let's start by distributing on the right side.
1. Distribute \( 32 \):
\[
32(4w - 4) = 128w - 128
\]
So the inequality becomes:
\[
18 > 128w - 128
\]
2. Next, add \( 128 \) to both sides to isolate the term with \( w \):
\[
18 + 128 > 128w
\]
Simplifying the left side gives:
\[
146 > 128w
\]
3. Now, divide both sides by \( 128 \):
\[
\frac{146}{128} > w
\]
Simplifying \( \frac{146}{128} \):
\[
\frac{146 \div 2}{128 \div 2} = \frac{73}{64}
\]
This can also be expressed as a decimal:
\[
73 \div 64 \approx 1.140625
\]
4. Thus, we can write the solution as:
\[
w < \frac{73}{64}
\]
Now, we check the provided answers. Since \(\frac{73}{64}\) is approximately equal to 1.14, the relevant answer from the options would be:
**w < 108**
However, this isn't the most precise answer, but that statement is true since \(\frac{73}{64}\) is less than \(108\).
The closest and correct interpretation from the options given is:
**w < 4** is incorrect based on our solution.
The solution should rather imply:
- The correct interpretation should be that **w < \frac{146}{128}** is the most accurate.
If we assume the options are referring to decimal approximations, \( w < 4 \) is indeed taking the higher limit instead of making a point of precision available.
So, the correct conclusion remains \( w < \frac{73}{64} \) or approximated to \( w < 1.14 \) and none exactly matches the options, the closest approximation though was \( \text{w < 4}. \)
But to plainly conclude:
- The inequality does yield **w < 4** seems to be true, but in real maths strictly, it was computed less than \( \frac{73}{64} \) so focus on that critical better approximation.
1. Distribute \( 32 \):
\[
32(4w - 4) = 128w - 128
\]
So the inequality becomes:
\[
18 > 128w - 128
\]
2. Next, add \( 128 \) to both sides to isolate the term with \( w \):
\[
18 + 128 > 128w
\]
Simplifying the left side gives:
\[
146 > 128w
\]
3. Now, divide both sides by \( 128 \):
\[
\frac{146}{128} > w
\]
Simplifying \( \frac{146}{128} \):
\[
\frac{146 \div 2}{128 \div 2} = \frac{73}{64}
\]
This can also be expressed as a decimal:
\[
73 \div 64 \approx 1.140625
\]
4. Thus, we can write the solution as:
\[
w < \frac{73}{64}
\]
Now, we check the provided answers. Since \(\frac{73}{64}\) is approximately equal to 1.14, the relevant answer from the options would be:
**w < 108**
However, this isn't the most precise answer, but that statement is true since \(\frac{73}{64}\) is less than \(108\).
The closest and correct interpretation from the options given is:
**w < 4** is incorrect based on our solution.
The solution should rather imply:
- The correct interpretation should be that **w < \frac{146}{128}** is the most accurate.
If we assume the options are referring to decimal approximations, \( w < 4 \) is indeed taking the higher limit instead of making a point of precision available.
So, the correct conclusion remains \( w < \frac{73}{64} \) or approximated to \( w < 1.14 \) and none exactly matches the options, the closest approximation though was \( \text{w < 4}. \)
But to plainly conclude:
- The inequality does yield **w < 4** seems to be true, but in real maths strictly, it was computed less than \( \frac{73}{64} \) so focus on that critical better approximation.