Solving a Linear Inequality:  Problem Type 4

Solving a linear inequality involves exactly the same steps as solving an equation with the addition of one step.  This additional step involves reversing the inequality symbol if you multiply or divide both sides of the inequality by a negative number.

This is required because multiplying by a negative changes the original sign.

For example:  -1 times -5 changes the negative five to a positive 5.

                        -1 times 7 changes the positive 7 to negative 7.

When the signs of numbers change, the relationship between the numbers changes.

For example:   2 < 5 but -2 > -5

                        -3 > -8 but 3 < 8

                        -4 < 5 but 4 > -5

                        2 > -3 but -2 < 3

This is why we reverse the inequality symbol when we multiply or divide both sides by a negative.  We know this process changes the signs of both sides, and we know this will change the relationship between the numbers…even if we have variables and don’t know exactly what the numbers are.

For example, if x > -5, then it must be the case that x < 5.

Problem 4 inequalities involve fractions.  We get rid of the fractions the same way we do when we have them in equations…multiply both sides by the least common denominator.  That will leave us with an inequality that does not have fractions which should make it easier to solve.

Example 1:  Solve

The only fraction is negative one half so we multiply both sides of the inequality by the denominator, 2.

Example 2:  Solve 

This problem has two fractions.  Their least common denominator is 6 so we multiply both sides of the inequality by 6 to get

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