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

Answer 1

To solve an inequality, I can provide a general example. Let's solve a common inequality: \(2x - 5 < 3\).

Step 1: Solve Algebraically

1. Add 5 to both sides: \[ 2x - 5 + 5 < 3 + 5 \] \[ 2x < 8 \]

2. Divide by 2: \[ x < 4 \]

Step 2: Interval Notation

The solution to the inequality \(x < 4\) in interval notation is: \[ (-\infty, 4) \]

Step 3: Graphically

To represent this on a number line:

1. Draw a horizontal line and mark a point for 4 on it.
2. Since the inequality is strict (\(<\)), use an open circle at 4 to indicate that 4 is not included in the solution.
3. Shade the line to the left of 4 to indicate that all numbers less than 4 are included in the solution.

Here's how the number line would look:

<---|---|---|---|---|---|---|---|---|---|---|---|---|--->
   -5  -4  -3  -2  -1   0   1   2   3   4   5   6   
                (o)

The point labeled (o) indicates an open circle at 4, with shading extending leftwards toward negative infinity.

Conclusion

The solution to the inequality \(2x - 5 < 3\) is:

• Algebraically: \(x < 4\)
• Interval notation: \((- \infty, 4)\)
• Graphically: A number line showing shading to the left of the open circle at 4.