Which are correct representations of the inequality –3(2x – 5) < 5(2 – x)? Select two options.

Question

x < 5
–6x – 5 < 10 – x
–6x + 15 < 10 – 5x
A number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right.
A number line from negative 3 to 3 in increments of 1. An open circle is at negative 5 and a bold line starts at negative 5 and is pointing to the left.

Answer 1

To solve the inequality \(-3(2x - 5) < 5(2 - x)\), let's first distribute both sides:

\[ -6x + 15 < 10 - 5x \]

Next, we will rearrange the inequality to isolate the variable \(x\). First, add \(5x\) to both sides:

\[ -6x + 5x + 15 < 10 \] \[ -x + 15 < 10 \]

Then, subtract 15 from both sides:

\[ -x < -5 \]

Now, multiply both sides by -1, which reverses the inequality:

\[ x > 5 \]

Now let's evaluate the choices based on our solution.

1. Inequalities:

   • \( x < 5\) is incorrect because our solution is \(x > 5\).
   • \( -6x - 5 < 10 - x \) is not equivalent to our derived inequality.
   • \( -6x + 15 < 10 - 5x \) is correct as it represents the original inequality.

2. Number Lines:

   • The first number line describes an open circle at 5 pointing to the right, which represents \(x > 5\) and is correct.
   • The second number line describes an open circle at \(-5\) pointing to the left, which is incorrect.

Thus, the correct choices are:

1. \(-6x + 15 < 10 - 5x\)
2. The number line indicating \(x > 5 (open circle at 5 with a line going to the right)\)