Question

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

Answer 1

To solve the inequality \(\frac{2}{3}x - \frac{1}{2}x > \frac{3}{4}\), we first need to combine the terms on the left side.

We can do this by finding a common denominator for \(\frac{2}{3}\) and \(\frac{1}{2}\). The least common multiple of 3 and 2 is 6.

1. Rewrite \(\frac{2}{3}x\) using the common denominator:
\[
\frac{2}{3}x = \frac{4}{6}x
\]

2. Rewrite \(\frac{1}{2}x\) using the common denominator:
\[
\frac{1}{2}x = \frac{3}{6}x
\]

3. Now we can combine the terms:
\[
\frac{4}{6}x - \frac{3}{6}x = \frac{1}{6}x
\]

Now, substitute this back into the inequality:
\[
\frac{1}{6}x > \frac{3}{4}
\]

4. To isolate \(x\), multiply both sides of the inequality by 6:
\[
x > 6 \cdot \frac{3}{4}
\]

5. Calculate \(6 \cdot \frac{3}{4}\):
\[
6 \cdot \frac{3}{4} = \frac{18}{4} = \frac{9}{2} = 4.5
\]

So the inequality simplifies to:
\[
x > 4.5
\]

### Graphing the solution on a real number line:
- Represent this solution on a number line:
- Use an open circle at \(4.5\) to indicate that \(4.5\) is not included in the solution.
- Shade the region to the right of \(4.5\) to represent all values greater than \(4.5\).

Here's how the graph would look:

```
<----(----|-----|-----|-----|-----|------>
0 1 2 3 4 4.5 5
↑
(not included)
```

In summary, the solution to the inequality \(\frac{2}{3}x - \frac{1}{2}x > \frac{3}{4}\) is \(x > 4.5\).