Solve the inequalities 2/3x -1/2x>3/4 and graph the solution set on real number line

Answer 1

To solve the inequality \( \frac{2}{3}x - \frac{1}{2}x > \frac{3}{4} \), let's first find a common denominator to combine the terms on the left.

1. The common denominator for \( \frac{2}{3} \) and \( \frac{1}{2} \) is 6. Convert both fractions:

   \[ \frac{2}{3}x = \frac{4}{6}x \] \[ \frac{1}{2}x = \frac{3}{6}x \]

2. Now substitute these back into the inequality:

   \[ \frac{4}{6}x - \frac{3}{6}x > \frac{3}{4} \]

3. Simplifying the left side gives:

   \[ \frac{1}{6}x > \frac{3}{4} \]

4. To isolate \( x \), multiply both sides by 6 (note that since 6 is a positive number, the inequality direction remains the same):

   \[ x > 6 \cdot \frac{3}{4} \] \[ x > \frac{18}{4} \] \[ x > \frac{9}{2} \] \[ x > 4.5 \]

The solution to the inequality is:

\[ x > 4.5 \]

Graphing the Solution Set

To graph this on the real number line:

1. Draw a number line.
2. Locate the point \( 4.5 \) on the line.
3. Since \( x \) must be greater than \( 4.5 \), place an open circle on \( 4.5 \) (indicating that \( 4.5 \) is not included in the solution).
4. Shade the line to the right of \( 4.5 \) to represent all numbers greater than \( 4.5 \).

This is how the graph will look:

<---|---|---|---|---|---|---|---|---|---|---|--->
    0   1   2   3   4   5   6   7   8   9  10
            (4.5)  ↑
                   |
  Solution:      x > 4.5

The shaded region represents the solution set to the inequality.