Solve the inequality:
2/3(3x-9) < 8
Use the distributive law to get rid of the parentheses.
2/3(3x-9) < 8
(2/3)(3x) + (2/3)(-9) < 8
2x + (-6) < 8
2x - 6 < 8
Add 6 to both sides
2x < 14
Divide both sides by 2
x < 7
That may be all your teacher wants. But sometimes
they want you to do more.
If you are asked for the solution set than that is
{x|x is real and x < 7}
The way that is read is:
"The set of all x such that x is a real number which
is less than or equal to seven".
If you are asked for more, i.e., then read the rest.
Draw the graph on a number line, and/or give the
solution in set-builder notation as well as in interval
notation.
The graph on a number line is done this way:
First we draw a number line:
—————————————————————————————————————————————
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11
Then we draw a circle at 7, but we have to go by this rule
to determine whether we will use an open circle like this "o"
or a closed circle like this "l". The rule is if it's " < " or
" > " then we use an open circle "o" and if its > or <,
then we use l. So since this is a case of <, we use "l"
———————————————————————————————l—————————————
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11
Now we use this rule:
If it's > or > then you shade to the right because it's
"greater than" and the greater numbers are to the right.
If it's < or < then you shade to the left because it's
"less than" and the lesser numbers are to the left.
So we shade to the right:
———————————————————————————————lnnnnnnnnnnnnn>
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11
That's the graph. The interval notation is
[7,¥)
If it had been x > 7, then the graph would have been this:
———————————————————————————————onnnnnnnnnnnnn>
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11
and the interval notation would have been
(7,¥)
Edwin
AnlytcPhil@aol.com