Question

solving one-variable inequalities
which are correct representations of the inequality $-3(2x - 5) < 5(2 - x)$? choose two correct answers.
$x < 5$
(number line with arrow pointing left from 5)
$-6x + 15 < 10 - 5x$
(number line with arrow pointing right from 5)
$-6x - 5 < 10 - x$

Explanation:

Step1: Expand both sides of the inequality

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

Step2: Solve the inequality for \(x\)

Add \(6x\) to both sides of the inequality \(-6x + 15<10 - 5x\):
\(-6x+6x + 15<10-5x + 6x\)
Simplify both sides: \(15<10 + x\)
Subtract 10 from both sides: \(15 - 10<10 + x-10\)
Simplify: \(5 5\)

The number line representation of \(x>5\) is a number line with an open circle at \(x = 5\) and an arrow pointing to the right (since \(x\) is greater than 5).

Now let's analyze the options:

• The option \(x < 5\) is incorrect because we found \(x>5\).
• The number line with the arrow pointing to the left (for \(x < 5\)) is incorrect.
• The inequality \(-6x + 15<10 - 5x\) (from step 1) is correct.
• The number line with an open circle at 5 and an arrow pointing to the right (representing \(x>5\)) is correct.
• The inequality \(-6x - 5<10 - x\) is incorrect because when we expanded \(-3(2x - 5)\) we should get \(-6x + 15\) not \(-6x-5\).

Answer:

The two correct answers are:

• \(-6x + 15<10 - 5x\)
• The number line with an open circle at \(x = 5\) and an arrow pointing to the right (representing \(x>5\))