Inequalities are solved similar to equations, with rules of inequality. For example, the solving of 3x+4<7 is as follows:
This can be graphed on a number line with a dot where the 1 is. For a greater-than or less-than inequality, an open dot(o) is used to represent the number. For a greater-than-or-equal-to or less-than-or-equal-to inequality, use a closed dot(•).
This would be graphed:
Note the open dot over 1.
In an inequality, one major difference is that when dividing or multiplying by a negative number, the inequality sign is switched.
Notice the flipped sign. When multiplying both sides by -3, the ≤ becomes ≥.
This would be graphed as:
Inequalities can be combined with and or or.
An and inequality has multiple inequality signs. For example:
1 < x < 10
Inequality operations can still be performed, but must affect all sections. When dividing or multiplying by a negative number, you must flip the signs.
And inequalities can also be graphed, by connecting two dots. For example, 1 < x < 10 is graphed:
An or inequality is two inequalities joined by an or
Which is graphed as two separate inequalities.
Interval notation is a way of describing inequalities as a set of numbers from a to b.
For example, x>3 would be described as:
Because it starts at 3 and goes to infinity.
A parenthesis is used for both sides because it approaches but never reaches 3 and infinity. If the inequality was x≤5 the interval notation would be:
Now a square bracket is by 5 because x can reach 5.
For or inequalities, two interval notation groups are used, separated by or.