Let's solve the inequality step by step. The original inequality is:
\[ 6x + 1 < -\frac{4x}{3} + 4. \]
Step 1: Eliminate the fraction
To eliminate the fraction, we can multiply every term by 3 (to clear the denominator):
\[ 3(6x + 1) < 3\left(-\frac{4x}{3} + 4\right). \]
This simplifies to:
\[ 18x + 3 < -4x + 12. \]
Step 2: Combine like terms
Now, let's move all the terms involving \( x \) to one side and the constant terms to the other side. Adding \( 4x \) to both sides gives:
\[ 18x + 4x + 3 < 12. \]
This simplifies to:
\[ 22x + 3 < 12. \]
Next, subtract 3 from both sides:
\[ 22x < 12 - 3, \] \[ 22x < 9. \]
Step 3: Solve for \( x \)
Now, divide both sides by 22:
\[ x < \frac{9}{22}. \]
Step 4: Write the solution in interval notation
The inequality indicates that \( x \) can take any value less than \( \frac{9}{22} \). In interval notation, this is expressed as:
\[ (-\infty, \frac{9}{22}). \]
Step 5: Graphing the solution on a number line
On a number line, we represent the solution as follows:
- Draw a number line.
- Mark the point \( \frac{9}{22} \).
- Use an open circle (or a hollow dot) at \( \frac{9}{22} \) to indicate that this value is not included in the solution.
- Shade the region to the left of \( \frac{9}{22} \) to represent all values less than \( \frac{9}{22} \).
Conclusion
So the solution to the inequality \( 6x + 1 < -\frac{4x}{3} + 4 \) is:
- Algebraically: \( x < \frac{9}{22} \)
- Graphically: represented as an open circle at \( \frac{9}{22} \) and shading to the left
- Interval notation: \( (-\infty, \frac{9}{22}) \)