Solve the inequality both algebraically and graphically. Draw a number line graph of the solution and give interval notation.

Question

3xplus1less thanminusStartFraction 6 x Over 5 EndFraction
plus3
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 21 EndFraction. The region to the right of a left bracket at StartFraction 10 Over 21 EndFraction is shaded.
0
1
0
B.
An infinite number line, labeled from 0 to 1, has tick marks in increments of StartFraction 1 Over 21 EndFraction. The region to the right of a left parenthesis at approximately 0.8 is shaded.
0
1
0
C.
An infinite number line, labeled from 0 to 1, has tick marks in increments of StartFraction 1 Over 21 EndFraction. The region to the left of a right parenthesis at StartFraction 10 Over 21 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 21 EndFraction. The region to the right of a left parenthesis at StartFraction 10 Over 21 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 21 EndFraction. The region to the left of a right parenthesis at approximately 0.8 is shaded.
0
1
0
F.
An infinite number line, labeled from 0 to 1, has tick marks in increments of StartFraction 1 Over 21 EndFraction. The region to the left of a right bracket at StartFraction 10 Over 21 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 21 EndFraction. The region to the left of a right bracket at approximately 0.8 is shaded.
0
1
0
H.
An infinite number line, labeled from 0 to 1, has tick marks in increments of StartFraction 1 Over 21 EndFraction. The region to the right of a left bracket at approximately 0.8 is shaded.
0
1
0

Answer 1

To solve the inequality
\[ 3x + 1 < -\frac{6x}{5} + 3, \] let's first manipulate it algebraically.

Step 1: Eliminate the fraction

To eliminate the fraction, we can multiply every term by 5 (the denominator):

\[ 5(3x + 1) < 5\left(-\frac{6x}{5} + 3\right). \]

This simplifies to:

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

Step 2: Combine like terms

Next, we'll get all the \(x\) terms on one side and the constant terms on the other:

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

which simplifies to:

\[ 21x < 10. \]

Step 3: Solve for \(x\)

Now, divide both sides by 21:

\[ x < \frac{10}{21}. \]

Step 4: Solution as an inequality

The solution in inequality form is: \[ x < \frac{10}{21}. \]

Part 2: Solution in interval notation

In interval notation, this can be expressed as: \[ (-\infty, \frac{10}{21}). \]

Part 3: Number line graph

To draw the number line graph, you would represent:

• Open interval to the left of \(\frac{10}{21}\) (not including \(\frac{10}{21}\)).
• The point \(\frac{10}{21}\) would be an open circle.

Choosing the correct graph

Since \( \frac{10}{21} \) is approximately \( 0.476 \), the graph that reflects this would represent all numbers less than \(\frac{10}{21}\), and since it's an open interval, we choose the graph corresponding to this.

Based on your descriptions for the options, the appropriate choice should be one that shades to the left of a point approximately \( 0.476 \) (where \( \frac{10}{21} \) is). Therefore, the correct choice is:

C. An infinite number line, labeled from 0 to 1, has tick marks in increments of \( \frac{1}{21} \). The region to the left of a right parenthesis at \( \frac{10}{21} \) is shaded.

In summary:

• Inequality: \( x < \frac{10}{21} \)
• Interval Notation: \( (-\infty, \frac{10}{21}) \)
• Graph: Option C.