Question

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Part 1
Solve the inequality both algebraically and graphically. Draw a number line graph of the solution and give interval notation.
6xplus1less thanminusStartFraction 4 x Over 3 EndFraction
plus4

Answer 1

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:

1. Draw a number line.
2. Mark the point \( \frac{9}{22} \).
3. Use an open circle (or a hollow dot) at \( \frac{9}{22} \) to indicate that this value is not included in the solution.
4. 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**: \( (-\infty, \frac{9}{22}) \)