Sovi.AI - AI Math Tutor

Scan to solve math questions
ENFRDEESJAZH-CNZH-TW

QUESTION IMAGE

this question has two parts. first, answer part a. then, answer part b.…

Question

this question has two parts. first, answer part a. then, answer part b.
part a
solve $-3(7n + 3) < 6n$. write the solution using set - builder notation.
part b
graph the solution on a number line.

Explanation:

Part A

Step1: Expand the left side

$$\begin{align*} -3(7n + 3)&<6n\\ -21n-9&<6n \end{align*}$$

Step2: Add \(21n\) to both sides

$$\begin{align*} -21n + 21n-9&<6n+21n\\ -9&<27n \end{align*}$$

Step3: Divide both sides by \(27\)

$$\begin{align*} \frac{-9}{27}&<\frac{27n}{27}\\ -\frac{1}{3}&

In set - builder notation, the solution is \(\{n|n>-\frac{1}{3}\}\)

Part B

Since \(n>-\frac{1}{3}\), the number line should have an open circle at \(-\frac{1}{3}\) (because the inequality is strict, i.e., \(n\) is not equal to \(-\frac{1}{3}\)) and the line should go to the right.

Looking at the options:

  • Option A has a closed circle (incorrect as the inequality is \(>\), not \(\geq\))
  • Option B has the line going to the left (incorrect as \(n>-\frac{1}{3}\))
  • Option C has an open circle at \(0\) (incorrect, the boundary is at \(n =-\frac{1}{3}\))
  • Option D is also incorrect as it has the wrong direction and wrong boundary representation.

If we assume a mis - labeling of the options (assuming the open circle is at \(-\frac{1}{3}\) and line to the right), the general rule for \(n>-\frac{1}{3}\) is an open circle at the boundary value and shading to the right.

Answer:

Part A: \(\{n|n>-\frac{1}{3}\}\)
Part B: (Assuming correct representation with open circle at \(-\frac{1}{3}\) and line to the right, but based on given options, there is likely a mis - label. If we consider the closest in terms of direction and open - circle concept, if we assume the open - circle is mis - placed in the options and we go by the rule: open circle for \(>\) and direction to the right)