## 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
$2 \lt x \lt 5$
.