Question

which are correct representations of the inequality $-3(2x - 5) < 5(2 - x)$? select two options.
$\square$ $x < 5$
$\square$ $-6x - 5 < 10 - x$
$\square$ $-6x + 15 < 10 - 5x$
$\square$ (number line with open circle at 5, arrow to the right)
$\square$ (number line with open circle at -5, arrow to the left)

Explanation:

Step1: Expand both sides

First, expand \(-3(2x - 5)\) and \(5(2 - x)\).
For \(-3(2x - 5)\), using the distributive property \(a(b - c)=ab - ac\), we get \(-3\times2x - (-3)\times5=-6x + 15\).
For \(5(2 - x)\), using the distributive property \(a(b - c)=ab - ac\), we get \(5\times2 - 5\times x = 10 - 5x\).
So the inequality becomes \(-6x + 15<10 - 5x\).

Step2: Solve the inequality for \(x\)

Start with \(-6x + 15<10 - 5x\).
Add \(6x\) to both sides: \(-6x + 6x+ 15<10 - 5x+ 6x\), which simplifies to \(15<10 + x\).
Subtract \(10\) from both sides: \(15 - 10<10 - 10+ x\), so \(5<x\) or \(x > 5\)? Wait, no, let's check again. Wait, when we add \(6x\) to both sides:
\(-6x + 15+6x<10 - 5x+6x\)
\(15<10 + x\)
Then subtract \(10\): \(15 - 10<x\)
\(5<x\) or \(x > 5\)? Wait, that's a mistake. Wait, original inequality after expanding is \(-6x + 15<10 - 5x\). Let's do it again.
Add \(5x\) to both sides: \(-6x+5x + 15<10 - 5x+5x\)
\(-x + 15<10\)
Subtract \(15\) from both sides: \(-x+15 - 15<10 - 15\)
\(-x<-5\)
Multiply both sides by \(-1\) (remember to reverse the inequality sign): \(x > 5\)? Wait, no, that can't be. Wait, let's re - expand:
\(-3(2x - 5)=-6x + 15\) (correct, because \(-3\times2x=-6x\) and \(-3\times(-5) = 15\))
\(5(2 - x)=10 - 5x\) (correct, \(5\times2 = 10\) and \(5\times(-x)=-5x\))
So the inequality is \(-6x + 15<10 - 5x\) (so the third option is correct).
Now, solve \(-6x + 15<10 - 5x\):
Add \(6x\) to both sides: \(15<10 + x\)
Subtract \(10\): \(5<x\) or \(x>5\). Wait, but the first option is \(x < 5\), that's wrong. Wait, maybe I made a mistake in solving.
Wait, let's do it again:
\(-6x + 15<10 - 5x\)
Subtract \(15\) from both sides: \(-6x<10 - 15 - 5x\)
\(-6x<-5 - 5x\)
Add \(5x\) to both sides: \(-6x + 5x<-5\)
\(-x<-5\)
Multiply both sides by \(-1\), reverse the inequality: \(x > 5\). Wait, but the number line with the open circle at \(5\) and arrow to the right (the fourth option) represents \(x>5\), and the third option \(-6x + 15<10 - 5x\) is the expanded form. Wait, but the first option is \(x < 5\), which is wrong. Wait, maybe the original problem has a typo? Wait, no, let's check the expansion again.
Wait, \(-3(2x - 5)\): distribute \(-3\) to \(2x\) and \(-5\): \(-3\times2x=-6x\), \(-3\times(-5)=15\), so \(-6x + 15\). Correct.
\(5(2 - x)\): \(5\times2 = 10\), \(5\times(-x)=-5x\), so \(10 - 5x\). Correct. So the inequality is \(-6x + 15<10 - 5x\) (third option is correct).
Now, solving \(-6x + 15<10 - 5x\):
\(-6x+5x<10 - 15\)
\(-x<-5\)
\(x > 5\) (multiply both sides by \(-1\), reverse inequality). So the number line with open circle at \(5\) and arrow to the right (the fourth option: the one with 5,6,7,...) represents \(x > 5\), and the third option \(-6x + 15<10 - 5x\) is correct. Wait, but the first option is \(x < 5\), which is incorrect. Wait, maybe I messed up the sign when distributing?
Wait, no, \(-3(2x - 5)=-6x + 15\) is correct. \(5(2 - x)=10 - 5x\) is correct. Then solving:
\(-6x + 15<10 - 5x\)
\(-6x+5x<10 - 15\)
\(-x<-5\)
\(x > 5\). So the correct options are \(-6x + 15<10 - 5x\) (third option) and the number line with open circle at 5 and arrow to the right (fourth option) and also, wait, maybe I made a mistake in the sign during solving. Wait, let's check with a test value. Let's take \(x = 6\) (which is greater than 5).
Left side of original inequality: \(-3(2\times6 - 5)=-3(12 - 5)=-3\times7=-21\)
Right side: \(5(2 - 6)=5\times(-4)=-20\)
Is \(-21<-20\)? Yes, that's true.
Take \(x = 4\) (less than 5):
Left side: \(-3(8 - 5)=-3\times3=-9\)
Right side: \(5(2 - 4)=5\times(-2)=-10\)
Is \(-9<-10\)? No, that's false. So \(x>5\) is correct. So the third option \(-6x + 15<10 - 5x\) and the fourth option (the number line with open circle at 5 and arrow to the right) and also, wait, the first option is \(x < 5\) which is wrong, the second option \(-6x - 5<10 - x\) is wrong (because the expansion of \(-3(2x - 5)\) is \(-6x + 15\), not \(-6x - 5\)). So the correct options are \(-6x + 15<10 - 5x\) (third option) and the number line with open circle at 5 and arrow to the right (fourth option). But the problem says "select two options", and among the options, \(-6x + 15<10 - 5x\) (third) and the number line with open circle at 5 and arrow to the right (fourth) are correct. Wait, but the first option is \(x < 5\) which is wrong, the second is wrong, the third is correct, the fourth is correct (represents \(x>5\)), and the fifth is wrong. So the two correct options are \(-6x + 15<10 - 5x\) and the number line with open circle at 5 and arrow to the right (the fourth option) and also, wait, maybe I made a mistake in the sign when solving. Wait, let's re - solve:
\(-6x + 15<10 - 5x\)
Add \(6x\) to both sides: \(15<10 + x\)
Subtract \(10\): \(5<x\), so \(x>5\). So the number line with open circle at 5 and arrow to the right (the fourth option: the one with 5,6,7 on the number line) is correct, and the third option \(-6x + 15<10 - 5x\) is correct.

Answer:

• \(-6x + 15 < 10 - 5x\)
• The number line with an open circle at 5 and the arrow pointing to the right (the fourth option in the list of options, the one with 5, 6, 7 marked on the number line)

(If we consider the options as per the checkboxes: the third option \(-6x + 15 < 10 - 5x\) and the fourth option (the number line with open circle at 5 and arrow to the right) are correct. Also, if we re - evaluate, maybe there was a miscalculation earlier. Wait, let's check the first option again. If we made a mistake in solving:

Wait, let's start over:

Inequality: \(-3(2x - 5)<5(2 - x)\)

Expand left side: \(-6x + 15\)

Expand right side: \(10 - 5x\)

So inequality: \(-6x + 15<10 - 5x\) (third option is correct)

Now, solve for \(x\):

\(-6x + 15<10 - 5x\)

Subtract \(15\) from both sides: \(-6x<10 - 15 - 5x\)

\(-6x<-5 - 5x\)

Add \(5x\) to both sides: \(-6x + 5x<-5\)

\(-x<-5\)

Multiply both sides by \(-1\), reverse inequality: \(x > 5\)

So the solution is \(x > 5\), which is represented by the number line with an open circle at 5 and arrow to the right (the fourth option: the one with 5,6,7,...) and the third option \(-6x + 15<10 - 5x\) is the expanded form. So the two correct options are:

• \(-6x + 15 < 10 - 5x\) (third checkbox)
• The number line with an open circle at 5 and arrow pointing to the right (fourth checkbox, the one with 5, 6, 7 on the number line)