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Simultaneous Inequalities

Sometimes it is required that we find a set of numbers that satisfy two inequalities simultaneously. Suppose we have to find  
\[x\]
  such that
\[5 \lt 3x-1 \lt 14\]

\[0 \lt 2x-2 \lt 10\]

From the first of these,
\[5+1 \lt 3x \lt 14+1 \rightarrow \frac{5+1}{3} \lt x \lt \frac{14+1}{3}\]
&
Hence
\[2 \lt x \lt 5\]

From the second,
\[0+2 \lt 2x \lt 10+2 \rightarrow \frac{0+2}{2} \lt x \lt \frac{10+2}{2}\]
&
Hence
\[1 \lt x \lt 6\]

\[x\]
  must satisfy both inequalities. From the first  
\[2 \lt x\]
  and from the second  
\[1 \lt x\]
. This last requirement is redundant since if  
\[2 \lt x\]
,  
\[1 \lt x\]
  is satisfied. Also, the are some numbers eg 1.5, that satisfy the second inequality but not the first, so do not satisfy both.
Similarly  
\[x \lt 5\]
  and  
\[x \lt 6\]
. The second of these is again redundant. The set of values of  
\[x\]
  that satisfy both inequalities is  
\[2 \lt x \lt 5\]
.