Question

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

Explanation:

Step1: Expand both sides

Expand \(-3(2x - 5)\) to get \(-6x + 15\), and expand \(5(2 - x)\) to get \(10 - 5x\). So the inequality becomes \(-6x + 15 < 10 - 5x\).

Step2: Solve for \(x\)

Subtract \(15\) from both sides: \(-6x < -5 - 5x\). Then add \(5x\) to both sides: \(-x < -5\). Multiply both sides by \(-1\) (and reverse the inequality sign): \(x > 5\)? Wait, no, wait. Wait, when we multiply or divide by a negative number, the inequality sign flips. Wait, let's do it again. Starting from \(-6x + 15 < 10 - 5x\). Subtract \(10\) from both sides: \(-6x + 5 < -5x\). Then add \(6x\) to both sides: \(5 < x\), which is \(x > 5\)? Wait, that contradicts the first option. Wait, maybe I made a mistake. Wait, let's re - expand. \(-3(2x - 5)\): using distributive property \(a(b - c)=ab - ac\), so \(-3\times2x-(-3)\times5=-6x + 15\). \(5(2 - x)=10 - 5x\). So the inequality is \(-6x + 15 < 10 - 5x\) (so the third option is correct). Now, let's solve for \(x\):
Subtract \(15\) from both sides: \(-6x<10 - 5x-15\), which simplifies to \(-6x < - 5-5x\).
Add \(5x\) to both sides: \(-6x + 5x<-5\), so \(-x < - 5\).
Multiply both sides by \(-1\): when we multiply an inequality by a negative number, the inequality sign flips. So \(x > 5\). Wait, but the first option is \(x < 5\), that's wrong. Wait, maybe I messed up the sign when expanding? Wait, no. Wait, let's check the number line. The fourth option has an open circle at \(5\) and the line going to the right (which represents \(x > 5\)), and the fifth option has an open circle at \(-5\) and line going to the left ( \(x < - 5\)). Wait, the third option is \(-6x + 15 < 10 - 5x\) which is the correct expansion. Then when we solve, we get \(x > 5\). So the correct options are the third one (\(-6x + 15 < 10 - 5x\)) and the fourth one (the number line with open circle at \(5\) and arrow to the right, representing \(x > 5\)). Wait, but the first option is \(x < 5\), which is wrong. Wait, maybe I made a mistake in solving. Let's do it again:

Starting with \(-3(2x - 5)<5(2 - x)\)

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

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

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

Now, subtract \(-6x + 15\) from both sides? No, better to get all \(x\) terms on one side. Let's add \(6x\) to both sides:

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

Simplify: \(15 < 10 + x\)

Subtract \(10\) from both sides: \(5 < x\), so \(x > 5\). So the number line that represents \(x > 5\) is the fourth option (open circle at \(5\), arrow to the right). And the third option is the correct expanded form. The first option is \(x < 5\) which is wrong. The second option: \(-6x - 5 < 10 - x\) is wrong because when we expanded \(-3(2x - 5)\) we should get \(-6x + 15\), not \(-6x - 5\). The fifth option is for \(x < - 5\), which is wrong. 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). Wait, but the problem says "select two options". Let's confirm the number line: the fourth option has an open circle at \(5\) and the line goes to the right, which is \(x > 5\), which is the solution. The third option is the correct expanded inequality.

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)