## 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 \[2 \lt x \lt 5\]

.