Question

Solve the inequality both algebraically and graphically. Draw a number line graph of the solution and give interval notation.
5xplus2less thanminusStartFraction 6 x Over 5 EndFraction
plus6
Question content area bottom
Part 1
The solution as an inequality is
  
enter your response here.
​(Type an inequality. Simplify your answer. Use integers or fractions for any numbers in the​ expression.)
Part 2
The solution in interval notation is
  
enter your response here.
​(Type your answer in interval notation. Use integers or fractions for any numbers in the​ expression.)
Part 3
Choose the correct graph of the solution below.
A.
An infinite number line, labeled from 0 to 1, has tick marks in increments of StartFraction 1 Over 31 EndFraction. The region to the left of a right parenthesis at approximately 0.7 is shaded.
0
1
0
B.
An infinite number line, labeled from 0 to 1, has tick marks in increments of StartFraction 1 Over 31 EndFraction. The region to the right of a left bracket at approximately 0.7 is shaded.
0
1
0
C.
An infinite number line, labeled from 0 to 1, has tick marks in increments of StartFraction 1 Over 31 EndFraction. The region to the left of a right bracket at StartFraction 20 Over 31 EndFraction is shaded.
0
1
0
D.
An infinite number line, labeled from 0 to 1, has tick marks in increments of StartFraction 1 Over 31 EndFraction. The region to the right of a left bracket at StartFraction 20 Over 31 EndFraction is shaded.
0
1
0
E.
An infinite number line, labeled from 0 to 1, has tick marks in increments of StartFraction 1 Over 31 EndFraction. The region to the right of a left parenthesis at approximately 0.7 is shaded.
0
1
0
F.
An infinite number line, labeled from 0 to 1, has tick marks in increments of StartFraction 1 Over 31 EndFraction. The region to the left of a right parenthesis at StartFraction 20 Over 31 EndFraction is shaded.
0
1
0
G.
An infinite number line, labeled from 0 to 1, has tick marks in increments of StartFraction 1 Over 31 EndFraction. The region to the left of a right bracket at approximately 0.7 is shaded.
0
1
0
H.
An infinite number line, labeled from 0 to 1, has tick marks in increments of StartFraction 1 Over 31 EndFraction. The region to the right of a left parenthesis at StartFraction 20 Over 31 EndFraction is shaded.
0
1
0

Answer 1

Let's solve the inequality step by step. The inequality we are given is:

\[ 5x + 2 < -\frac{6x}{5} + 6 \]

### Part 1: Solve Algebraically

1. Start by eliminating the fraction. We can multiply the entire inequality by 5 (which is positive, so the direction of the inequality remains the same):

\[ 5(5x + 2) < 5\left(-\frac{6x}{5} + 6\right) \]

This simplifies to:

\[ 25x + 10 < -6x + 30 \]

2. Next, combine like terms. Add \(6x\) to both sides:

\[ 25x + 6x + 10 < 30 \]

Which simplifies to:

\[ 31x + 10 < 30 \]

3. Subtract 10 from both sides:

\[ 31x < 20 \]

4. Divide by 31 (which is positive, so the direction of the inequality remains the same):

\[ x < \frac{20}{31} \]

So the solution as an inequality is:

\[ x < \frac{20}{31} \]

### Part 2: Interval Notation

In interval notation, this is expressed as:

\[ (-\infty, \frac{20}{31}) \]

### Part 3: Number Line Graph

To represent the solution on a number line, you would shade to the left of \(\frac{20}{31}\) pointing out that this is an open interval (not including \(\frac{20}{31}\)).

Now among the options provided:

- Since we are shading to the left of \(\frac{20}{31}\) with a right parenthesis (indicating that \(\frac{20}{31}\) is not included), the correct option would be:

**C. An infinite number line, labeled from 0 to 1, has tick marks in increments of StartFraction 1 Over 31 EndFraction. The region to the left of a right bracket at StartFraction 20 Over 31 EndFraction is shaded.**

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To summarize:

- **Part 1:**
\[ x < \frac{20}{31} \]

- **Part 2:**
\[ (-\infty, \frac{20}{31}) \]

- **Part 3 selection:**
**C**.