## Quadratic Inequalities

To solve the inequality\[x^2-4x+3 \gt 0\]

, first factorise to give \[(x-1)(x-3) \gt 0\]

.Now sketch the graph of

\[y=x^2-4x+3\]

.Because we are solving

\[x^2-4x+3 \gt 0\]

we want those values of \[y \gt 0\]

.From the graph we see this is true for

\[x \lt 0\]

or \[x \gt 3\]

.For the inequality

\[x^2-4x+3 \lt 0\]

, we need that part of the graph below the \[x\]

axis. The solution is \[1 \lt x \lt 3\]

.To solve the inequality

\[x^2-4x+3 \ge 0\]

, just replace each 'greater than' sign with 'greater than or equal to' and each 'less than' sign with a 'less than or equal to' sign. We get \[x \le 1\]

or \[x \ge 3\]

.