Answer
$(-∞,\frac{1}{2})∪ (\frac{7}{5}, ∞)$
Work Step by Step
Consider the Rational Inequality as follows:
$\frac{x+4}{2x-1}<3$
Here are the steps required for Solving Rational Inequalities:
Step 1: One side must be zero and the other side can have only one fraction, so we simplify the fractions if there is more than one fraction. Let us subtract 3 from both sides to obtain zero on the right.
$\frac{x+4}{2x-1}<3$
$\frac{x+4}{2x-1}-3<0$
$\frac{x+4-3(2x-1)}{2x-1}<0$
$\frac{-5x+7}{2x-1}<0$
Step 2: Critical or Key Values are first evaluated. In order to this, set the numerator and denominator of the fraction equal to zero and then simplified rational inequality is solved.
$-5x+7 = 0$
This implies
$x =7/5$
And
$2x-1=0$
This implies
$ x = 1/2$
These solutions are used as boundary points on a number line.
Step 3: Locate the boundary points on a number line found in Step 2 to divide the number line into intervals.
The boundary points divide the number line into three intervals:
$(-∞,\frac{1}{2}), (\frac{1}{2}, \frac{7}{5}), (\frac{7}{5}, ∞)$
Step 4: Now, one test value within each interval is chosen and $f$ is evaluated at that number.
Intervals: $(-∞,\frac{1}{2}), (\frac{1}{2}, \frac{7}{5}), (\frac{7}{5}, ∞)$
Test value: $0$ $1$ $1.5$
Sign Change: Negative Positive Negative
$f (x)< 0?$: T F T
Step 5: Write the solution set, selecting the interval or intervals that satisfy the given inequality. Solve $f (x)< 0 $.
Based on our work done in Step 4, we see that $f (x)< 0 $ for all x in
$(-∞,\frac{1}{2})$or$ (\frac{7}{5}, ∞)$
Conclusion: The interval notation of the given inequality is
$(-∞,\frac{1}{2})∪ (\frac{7}{5}, ∞)$ and the graph of the solution set on a number line is shown as follows:
