Solving inequalities is not so different from solving regular equations. In fact, an inequality sign (<,>,≤,≥) is treated the same as an equal (=) sign when solving inequalities involving only addition or subtraction. Before all that, let us define the different inequality signs:
- < indicates that the expression on the left is less than the expression
on the right, for example:
indicates that 3 is less than 9 and we all know that to be true.
means that the solution to the expression on the left is less than zero, but more on that later.
- > indicates that the expression on the left is greater than the
expression on the right, for example:
shows that 9 is greater than 3 and similarly
means that the value of 2x - 9 is greater than 11
- ≤ indicates that the value on the left is less than or equal to
the value on the right, for example:
means that the value of the expression on the left must be less than or equal to 25
- ≥ indicates that the value on the left is greater than or equal to
the value on the right, for example:
Inequalities are best thought of as representing different regions on a number line:
- < represents the region to the left of a given number, for example
represents all the numbers to the left of 3 (less than 3), which in on the number line below is shown by all the numbers in the direction of the arrow
- > represents the region to the right of a given number, for example
represents all the numbers to the right of 3 (greater than 3), which in on the number line below is shown by all the numbers in the direction of the arrow
- ≤ represents the region from a given number to the left of that same number,
represents all the numbers to the left of 5 (less than 5) including 5 itself as shown on the number line below
- ≥ represents the region from a given number to the right of that same
number, for example;
represents all the numbers to the left of 1 (less than 1) including 1 itself as shown on the number line below
Most linear inequalities can be solved just the same as linear equations: Addition and subtraction of any number (positive or negative) can be done to the expression on either side of the inequality without changing the inequality itself. In other words, it would be the same as in any ordinary equation.
For example; solve for x in
Subtract 9 from both sides of the inequality as follows:
and the answer would be
Observe that the above is solved in the same was as solving ordinary equations. This is because addition and subtraction have no effect on the inequality sign. Multiplication and division are different, however, as the inequality sign is treated differently depending on whether you are multiplying by a positive or negative number.
Multiplying or dividing through the inequality expression by a positive number has no effect on the inequality sign and is treated as you would an ordinary equation.
For example, solve for x in:
First add 9 to both sides of the inequality
then divide through by 3
which results in
Multiplying or dividing through the inequality by a negative number has the effect of reversing the inequality sign, for example from < to > as shown below
is solved as follows;
dividing by -2 reverses the inequality sign resulting in:
To prove why the above is true, let us first understand the answer:
means that x can take on any value as long as that value is greater than -4 and the original equation will be true. To prove this, let us try different values of x;
First, try x = 1, 1 is greater than -4 so substituting x = 1 in the original expression should give a mathematically correct inequality
which is true.
Next, let us try a value of x less than -4, for example, substitute x = -5
but 7 is not less than 5 which means that the solution we got as x > -4 is true. Try substituting different values of x into the expression -3 - 2x < 5 and no matter what value you choose, as long as x > -4 the solution should always hold true.
Solving Polynomial Inequalities
Solving more complicated polynomial inequalities is not so straightforward.
For example, solve the inequality below for x
If the above had been an equation, finding the roots by factoring or completing the square would be all thats necessary. However, inequalities are different. The above is solved as follows:
First factor the expression on the left
which means that either the solution to the expression on the left is x = -2 or x = 6 but this is not the end.
Next you have to test the different regions on the number line to find out exactly where the solution to the entire inequality lies. First test x < -2 by picking a number on the left of -2 on the number line and then substitute it into the original inequality i.e.
which is not true and so we conclude that x is not less than -2.
Next we test x > -2 and by picking a number on the right of -2 on the number line, but for the time being this number has to be less than 6
which is true and so we conclude that x > -2. But since we had 2 roots, we have to test x = 6 as well.
We don't have to test for x < 6 since any number less than 6 is also greater than -2 and we already proved that x > -2. So we test for x > 6
which is not true, 9 is not less than zero and so the x is not greater than 6. This implies that x < 6
And thus the solution to x^2 - 4x - 12 < 0 can be given as:
which means that x lies on the region on the number line between -2 and +6