Question

1. what is the solution \\(\frac{3}{8}x < \frac{2}{6}\\) ?
2. what is the solution \\(\frac{x}{9} - \frac{6}{9} > \frac{2}{9}\\) ?
3. find the solution and graph \\(-3 < -\frac{x}{3}\\)

(number line with -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 marked)

Explanation:

Problem 6:

Step1: Solve for \( x \) by dividing both sides by \( \frac{3}{8} \) (or multiplying by \( \frac{8}{3} \))

To solve \( \frac{3}{8}x < \frac{2}{6} \), we multiply both sides by the reciprocal of \( \frac{3}{8} \), which is \( \frac{8}{3} \). Since \( \frac{3}{8} \) is positive, the inequality sign remains the same.
\[
x < \frac{2}{6} \times \frac{8}{3}
\]

Step2: Simplify the right - hand side

First, simplify \( \frac{2}{6}=\frac{1}{3} \). Then \( \frac{1}{3}\times\frac{8}{3}=\frac{8}{9} \)
\[
x < \frac{8}{9}
\]

Step1: Add \( \frac{6}{9} \) to both sides of the inequality

To solve \( \frac{x}{9}-\frac{6}{9}>\frac{2}{9} \), we add \( \frac{6}{9} \) to both sides.
\[
\frac{x}{9}-\frac{6}{9}+\frac{6}{9}>\frac{2}{9}+\frac{6}{9}
\]

Step2: Simplify both sides

The left - hand side simplifies to \( \frac{x}{9} \), and the right - hand side simplifies to \( \frac{2 + 6}{9}=\frac{8}{9} \). So we have \( \frac{x}{9}>\frac{8}{9} \)

Step3: Multiply both sides by 9

Since 9 is positive, the inequality sign remains the same.
\[
x>8
\]

Step1: Multiply both sides by - 3

To solve \( - 3<-\frac{x}{3} \), we multiply both sides by - 3. When we multiply or divide an inequality by a negative number, the inequality sign flips.
\[
(-3)\times(-3)>x
\]

Step2: Simplify the left - hand side

\( (-3)\times(-3) = 9 \), so the inequality becomes \( 9>x \) or \( x < 9 \)

Graphing:

• The number line has marks from - 3 to 12.
• We draw an open circle at 9 (because \( x < 9 \), not \( x\leq9 \)) and shade the region to the left of 9.

Answer:

\( x < \frac{8}{9} \)

Problem 7: