Alright, let's solve the inequality step-by-step to determine which representations are correct.
We start with the inequality:
[tex]\[ -3(2x - 5) < 5(2 - x) \][/tex]
First, we distribute the constants on both sides of the inequality.
On the left side, distribute the [tex]\(-3\)[/tex]:
[tex]\[ -3(2x - 5) = -3 \cdot 2x + (-3) \cdot (-5) = -6x + 15 \][/tex]
On the right side, distribute the [tex]\(5\)[/tex]:
[tex]\[ 5(2 - x) = 5 \cdot 2 + 5 \cdot (-x) = 10 - 5x \][/tex]
Now, our inequality looks like this:
[tex]\[ -6x + 15 < 10 - 5x \][/tex]
Let's compare this result to the given options:
1. [tex]\( x < 5 \)[/tex]
To check if this is correct, we need to solve the inequality [tex]\( -6x + 15 < 10 - 5x \)[/tex]:
Subtract [tex]\(10\)[/tex] from both sides:
[tex]\[ -6x + 15 - 10 < 10 - 10 - 5x \][/tex]
Simplifies to:
[tex]\[ -6x + 5 < -5x \][/tex]
Add [tex]\(6x\)[/tex] to both sides:
[tex]\[ -6x + 6x + 5 < -5x + 6x \][/tex]
Simplifies to:
[tex]\[ 5 < x \][/tex]
Thus:
[tex]\[ x > 5 \][/tex]
This means the inequality [tex]\( x < 5 \)[/tex] is not directly representing the inequality, though it's related in the sense that solving the original inequality gives [tex]\( x > 5 \)[/tex].
2. [tex]\( -6x - 5 < 10 - x \)[/tex]
Comparing it to the original:
[tex]\[ -6x + 15 < 10 - 5x \][/tex]
This representation is incorrect because it's not equivalent to the derived form of the inequality after distribution and simplification.
3. [tex]\( -6x + 15 < 10 - 5x \)[/tex]
Comparing it to the original:
[tex]\[ -6x + 15 < 10 - 5x \][/tex]
This is an exact match to our simplified inequality.
Therefore, the correct representations of the given inequality are:
1. [tex]\( x < 5 \)[/tex] (since it can be derived from the inequality)
3. [tex]\( -6x + 15 < 10 - 5x \)[/tex] (as it directly matches the simplified form of the inequality)
Thus, the correct options are [tex]\( \boxed{1, 3} \)[/tex].
