Question

Linear graph line shows solid red circle on 5 and shaded to the left.
3/7×+1/7<(underscore < ) 1 6/7
2/5× -4/5>(>underscore) 1 1/5
3/5×+ 2/5>(underscore)3 2/5
1/5×- 3/5<(underscore) 2/5

Answer 1

To solve the inequalities represented by the given statements on the graph, let's break each one down.

### 1. \( \frac{3}{7}x + \frac{1}{7} < 1 \frac{6}{7} \)

First, convert \( 1 \frac{6}{7} \) to an improper fraction:
\[
1 \frac{6}{7} = \frac{7}{7} + \frac{6}{7} = \frac{13}{7}
\]
Now, solve the inequality:
\[
\frac{3}{7}x + \frac{1}{7} < \frac{13}{7}
\]
Subtract \( \frac{1}{7} \) from both sides:
\[
\frac{3}{7}x < \frac{13}{7} - \frac{1}{7} = \frac{12}{7}
\]
Now, multiply both sides by \( \frac{7}{3} \):
\[
x < \frac{12}{7} \times \frac{7}{3} = \frac{12}{3} = 4
\]
So, the solution is:
\[
x < 4
\]

### 2. \( \frac{2}{5}x - \frac{4}{5} > 1 \frac{1}{5} \)

Convert \( 1 \frac{1}{5} \) to an improper fraction:
\[
1 \frac{1}{5} = \frac{5}{5} + \frac{1}{5} = \frac{6}{5}
\]
Now, solve the inequality:
\[
\frac{2}{5}x - \frac{4}{5} > \frac{6}{5}
\]
Add \( \frac{4}{5} \) to both sides:
\[
\frac{2}{5}x > \frac{6}{5} + \frac{4}{5} = \frac{10}{5} = 2
\]
Now, multiply both sides by \( \frac{5}{2} \):
\[
x > 2 \times \frac{5}{2} = 5
\]
So, the solution is:
\[
x > 5
\]

### 3. \( \frac{3}{5}x + \frac{2}{5} > 3 \frac{2}{5} \)

Convert \( 3 \frac{2}{5} \) to an improper fraction:
\[
3 \frac{2}{5} = \frac{15}{5} + \frac{2}{5} = \frac{17}{5}
\]
Now, solve the inequality:
\[
\frac{3}{5}x + \frac{2}{5} > \frac{17}{5}
\]
Subtract \( \frac{2}{5} \) from both sides:
\[
\frac{3}{5}x > \frac{17}{5} - \frac{2}{5} = \frac{15}{5} = 3
\]
Now, multiply both sides by \( \frac{5}{3} \):
\[
x > 3 \times \frac{5}{3} = 5
\]
So, the solution is:
\[
x > 5
\]

### 4. \( \frac{1}{5}x - \frac{3}{5} < \frac{2}{5} \)

Now, solve the inequality:
\[
\frac{1}{5}x - \frac{3}{5} < \frac{2}{5}
\]
Add \( \frac{3}{5} \) to both sides:
\[
\frac{1}{5}x < \frac{2}{5} + \frac{3}{5} = \frac{5}{5} = 1
\]
Now, multiply both sides by \( 5 \):
\[
x < 5
\]
So, the solution is:
\[
x < 5
\]

### Summary of Solutions
1. \( x < 4 \)
2. \( x > 5 \)
3. \( x > 5 \)
4. \( x < 5 \)

These inequalities represent different ranges for the variable \( x \). As such, the solution sets can be visualized on a number line, where shaded regions indicate where the solutions lie.