Consider the inequality ‘$\,x < x + 1\,$’.
Let
$\,x\,$ be any real number.
Then,
$\,x+1\,$ lies one unit to the right of $\,x\,$ on a number line.
Therefore,
$\,x\,$ always lies to the left of $\,x+1\,$,
so the sentence ‘$\,x < x + 1\,$’ is always true.
Consider the inequality ‘$\,x -1> x\,$’.
Let
$\,x\,$ be any real number.
Then,
$\,x-1\,$ lies one unit to the left of $\,x\,$ on a number line.
Therefore,
$\,x-1\,$ never lies to the right of $\,x\,$,
so the sentence ‘$\,x-1 > x\,$’ is always false.
For these exercises, you should think in terms of position on a number line,
as in the previous examples.
Determine if each inequality is ALWAYS TRUE or ALWAYS FALSE.