Find the solution set to the system of inequalities
\begin{equation*}
\begin{aligned}
x^2 - 4x &\geq -3 \\
\frac{x-5}{x-7} &\lt 0
\end{aligned}
\end{equation*}
Solution.
The first inequality is equivalent to \(x^2 - 4x + 3 \geq 0\text{.}\) The second one is already in a form where it is with respect to 0.
Since \(x^2 - 4x + 3 = (x - 3)(x-1)\text{,}\) the first expression equals 0 when \(x = 1\) or when \(x = 3\text{.}\) The expression \(\frac{x-5}{x-7}\) equals zero when \(x=5\text{,}\) and is undefined when \(x = 7\text{.}\) Thus the set of points that partition the remainder of the real line into open intervals is \(\{1, 3, 5, 7\}\text{.}\)
We need to test a point in each of the five intervals in the partition, and then be careful about the endpoints of the intervals.
For the interval \((-\infty, 1)\) we pick \(x = 0\) as the test point. The first inequality does not hold when \(x = 0\text{,}\) so no point in \((-\infty, 1)\) belongs to the solution set. The second inequality does not hold when \(x = 1\text{,}\) so this point is not in the solution set.
For \((1, 3)\text{,}\) we pick \(x = 2\) as the test point. The first inequality does not hold when \(x = 2\text{,}\) so no point in \((1, 3)\) belongs to the solution set.
For the interval \((3, 5)\text{,}\) we choose \(x = 4\) as the test point. The first inequality holds when \(x = 4\text{,}\) but the second one does not, so no point in \((3, 5)\) belongs to the solution set. The second inequality does not hold when \(x = 5\text{,}\) so this point is not in the solution set.
For the interval \((5, 7)\text{,}\) we choose \(x = 6\) as the test point. Both inequalities hold when \(x = 6\text{,}\) so all points in \((5, 7)\) belong to the solution set. The second inequality is undefined at \(x = 7\text{,}\) so this point is not in the solution set.
For the interval \((7, \infty)\text{,}\) we choose \(x = 8\) as the test point. The first equality holds when \(x = 8\text{,}\) but the second one does not, so no point in \((7, \infty)\) belongs to the solution set.
The solution set to the system of inequalities is therefore \((5, 7)\text{.}\) Its graph on the real line is shown below.