Question
which set expresses the solution to the compound inequality? -2 ≤ 2x+1/3 -4 < 1?
A) {x : 3 ≤ x < 7}
B) {x : 5/2 ≤ x < 1}
C) {x : 5/2 ≤ x < 1}
D) {x : -3/2 ≤ x < 1}
A) {x : 3 ≤ x < 7}
B) {x : 5/2 ≤ x < 1}
C) {x : 5/2 ≤ x < 1}
D) {x : -3/2 ≤ x < 1}
Answers
GPT-4o mini
To solve the compound inequality \(-2 \leq \frac{2x + 1}{3} - 4 < 1\), we will break it down into two parts.
First, let's address the left side of the compound inequality:
1) \(-2 \leq \frac{2x + 1}{3} - 4\)
Add 4 to both sides:
\[
-2 + 4 \leq \frac{2x + 1}{3}
\]
This simplifies to:
\[
2 \leq \frac{2x + 1}{3}
\]
Next, multiply both sides by 3:
\[
6 \leq 2x + 1
\]
Subtract 1 from both sides:
\[
5 \leq 2x
\]
Now divide by 2:
\[
\frac{5}{2} \leq x
\]
This can also be expressed as:
\[
x \geq \frac{5}{2}
\]
Now let's address the right side of the compound inequality:
2) \(\frac{2x + 1}{3} - 4 < 1\)
Add 4 to both sides:
\[
\frac{2x + 1}{3} < 5
\]
Multiply both sides by 3:
\[
2x + 1 < 15
\]
Subtract 1 from both sides:
\[
2x < 14
\]
Now divide by 2:
\[
x < 7
\]
Now we have the solutions from both parts:
\[
\frac{5}{2} \leq x < 7
\]
In set notation, this is expressed as:
\[
\{ x : \frac{5}{2} \leq x < 7 \}
\]
Now we need to find the correct answer from the given options. The answer is closest to:
B) \(\{ x : \frac{5}{2} \leq x < 1 \}\)
However, notice that it looks like there's a typo in the answer options because \(< 1\) is not correct based on our derived inequalities. The correct condition should state \(< 7\), which is not present.
Therefore, the answer depends on clarifying your available options, but based on the original compilation provided, the correct solution is:
\(\{ x : \frac{5}{2} \leq x < 7 \}. \)
If we go strictly by the available options, there's no completely correct one. If we were to pick from these options strictly on the available constructions, and given that B is repeated, this might be a copy error.
Thus, I can only say none of the provided options strictly represents the solution. Please check the options again, as the correct derived solution doesn't match the available selections.
First, let's address the left side of the compound inequality:
1) \(-2 \leq \frac{2x + 1}{3} - 4\)
Add 4 to both sides:
\[
-2 + 4 \leq \frac{2x + 1}{3}
\]
This simplifies to:
\[
2 \leq \frac{2x + 1}{3}
\]
Next, multiply both sides by 3:
\[
6 \leq 2x + 1
\]
Subtract 1 from both sides:
\[
5 \leq 2x
\]
Now divide by 2:
\[
\frac{5}{2} \leq x
\]
This can also be expressed as:
\[
x \geq \frac{5}{2}
\]
Now let's address the right side of the compound inequality:
2) \(\frac{2x + 1}{3} - 4 < 1\)
Add 4 to both sides:
\[
\frac{2x + 1}{3} < 5
\]
Multiply both sides by 3:
\[
2x + 1 < 15
\]
Subtract 1 from both sides:
\[
2x < 14
\]
Now divide by 2:
\[
x < 7
\]
Now we have the solutions from both parts:
\[
\frac{5}{2} \leq x < 7
\]
In set notation, this is expressed as:
\[
\{ x : \frac{5}{2} \leq x < 7 \}
\]
Now we need to find the correct answer from the given options. The answer is closest to:
B) \(\{ x : \frac{5}{2} \leq x < 1 \}\)
However, notice that it looks like there's a typo in the answer options because \(< 1\) is not correct based on our derived inequalities. The correct condition should state \(< 7\), which is not present.
Therefore, the answer depends on clarifying your available options, but based on the original compilation provided, the correct solution is:
\(\{ x : \frac{5}{2} \leq x < 7 \}. \)
If we go strictly by the available options, there's no completely correct one. If we were to pick from these options strictly on the available constructions, and given that B is repeated, this might be a copy error.
Thus, I can only say none of the provided options strictly represents the solution. Please check the options again, as the correct derived solution doesn't match the available selections.