Solve the inequality both algebraically and graphically. Give the solution in interval notation and draw it on a number line graph.

Question

StartFraction x minus 3 Over 4 EndFraction
less thaneight thirds

Answer 1

To solve the inequality \(\frac{x - 3}{4} < \frac{8}{3}\), we can follow these steps.

Algebraic Solution:

1. Multiply both sides by 4 (to eliminate the fraction). Remember that since 4 is positive, the direction of the inequality remains unchanged:

   \[ x - 3 < \frac{8}{3} \cdot 4 \]

   Calculating \(\frac{8}{3} \cdot 4\):

   \[ \frac{8 \cdot 4}{3} = \frac{32}{3} \]

   So, the inequality now becomes:

   \[ x - 3 < \frac{32}{3} \]

2. Add 3 to both sides:

   \[ x < \frac{32}{3} + 3 \]

   Convert 3 into thirds:

   \[ 3 = \frac{9}{3} \]

   Now the right-hand side becomes:

   \[ \frac{32}{3} + \frac{9}{3} = \frac{32 + 9}{3} = \frac{41}{3} \]

Thus, the solution in interval notation is:

\[ (-\infty, \frac{41}{3}) \]

Graphical Solution:

To represent this solution graphically on a number line:

1. Draw a horizontal line to represent the number line.
2. Mark the point \(\frac{41}{3}\) (approximately 13.67) on the number line.
3. Since the inequality is strict (\(<\)), use an open circle at \(\frac{41}{3}\) to indicate that this value is not included in the solution set.
4. Shade the area to the left of \(\frac{41}{3}\) to indicate that all numbers less than \(\frac{41}{3}\) are included in the solution.

Number Line Representation

Here's a simple representation:

<----(---●----------------)--->
      -∞    13.67    +∞

Where:

• The open circle \(\circ\) at \(\frac{41}{3}\) indicates that this point is not included.
• The line extending leftwards indicates all values less than \(\frac{41}{3}\).

Final Answer: The solution set in interval notation is \((-∞, \frac{41}{3})\).