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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  
\[x \ge 3\]
.

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