A1Side
Solving Linear Inequalities
(single variable)
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bullet Remember:
The process of solving a linear inequality is the same as solving a linear equation, except ...
... when you multiply (or divide) an inequality by a negative value,
you must reverse the direction of the inequality.

bullet Solving Linear Inequalities:

      For more basic problems on solving inequalities, see Refresher section.

ex1
ti84c
For calculator help with linear inequalities
click here.

Solve and graph the solution set of:   iqmath13
Dealing with Fractions:
Hint:
In this problem, the
negative signs appear in front of the two fractional terms. It may be easier to place the negative signs in the numerators when solving the problem, so as not to "forget" them. igmath14a

This example places the negative sign in the numerators.

Multiply all terms by the least common denominator (which is 6).

Add 4x to both sides.
The solution is easier if you move the smaller x value.

Be sure to write the 0 on the right side.

Subtract 12 from both sides.

Divide both sides by 3.

 

igmath14

circlegraph15

Other notations:
ex1notation

ex2 beware

Solve and graph the solution set of:   iqmath10
Reverse the Inequality Symbol:
Remember:
When you multiply or divide by a negative value (-1), reverse the direction of the inequality symbol.

Multiply both sides by the least common denominator, which is 8.


Divide both sides by -1, and flip the direction of the inequality.

Note: The direction of the inequality was reversed since we divided by a negative value (-1).

Graph using a closed circle for -2 (since x can equal -2) and an arrow to the left (since our symbol is less than or equal to).

iqmath10a

circlegraph12

Other notations:


ex3

Solve and graph the solution set of:   -5x + 5 > x - 30
Variable on the Right:
Remember:
If the variable ends up on the right side of the inequality, after being solved, don't panic! The variable on the right is perfectly OK. If you want to move the variable to the left side, be sure to reverse the direction of the inequality symbol. 6 > x is the same as x < 6

Move the x's to one side of the inequality symbol. (Add +5x to both sides.)

Add +30 to both sides to combine the constants (numbers).

Divide both sides by 6 to arrive at the answer.

Other notations:
ex1notation


ex3

Solve and graph the solution set of:    iqmath12
Dealing with ≠ (not equal):
If the problem deals with "not equal" (≠), solve the problem as if you were solving with an equal sign. That solution will be the only value that will make the "not equal" version FALSE. All other values will make it TRUE.

Subtract 3 from both sides.

Multiply both sides by 2.

The solution to this inequality is all real numbers except 2. The number 2 is the only value which makes this inequality false, because it makes both sides equal to 4.
iqmath12c

igmath12b


circlegraph14a


Other notations:

ex1notation



ex3 beware

Solve and graph the solution set of:    2(3x - 1) > 6x - 5
Dealing with ALL Answers TRUE:
If the variable cancels out when solved, and you are left with a TRUE numerical statement, the solution will be ALL REAL NUMBERS work.

Distribute across the parentheses.

Subtract 6x from both sides.

Notice the TRUE numerical result.

The solution to this inequality is the entire set (all) Real Numbers.

The graph is the entire number line.

2(3x - 1) > 6x - 5
6x - 2 > 6x - 5
-6x        -6x       
-2 > -5 TRUE


This tells us that any x-value that is a Real number will make this inequality true.

Solution: ALL REAL NUMBERS

ex3 beware

Solve and graph the solution set of:    4(2x + 1) < 8x + 4
Dealing with ALL Answers FALSE:
If the variable cancels out when solved, and you are left with a FALSE numerical statement, there will be NO solution.

Distribute across the parentheses.

Subtract 8x from both sides.

Notice the FALSE numerical result.

The solution to this inequality is the "empty set" (empty). There are no x-values which will make this inequality true.

iqmath11
This tells us that NO x-values will make this inequality true.

circlegraph13

There is nothing to be graphed.

In this problem, the left side of the inequality is simply another way of writing the right side of the inequality. The two sides are algebraically EQUAL to one another. Since a quantity can never be less than itself, this inequality is never true.

ex3 beware

Solve for x:    ax > 6a
Dealing with Variable Coefficients:
If the coefficient of the stated variable, x, is a "letter" (another variable), there will be NO solution UNLESS you are told (in the problem) that the "letter" coefficient is "positive" or "negative".

At first glance, it seems that dividing both sides by a will solve the problem.

BUT ...

We just don't know (in this problem) if a is a positive value or a negative value. If a is positive, our solution of x > 6 will be true.

But if a is negative, we need to reverse the direction of the inequality, and the solution is x < 6.

Since we were not told in the problem whether a is positive or negative, we cannot state a solution to this problem.
Therefore, there is NO SOLUTION!

positive_________________________________

negative

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reminder An inequality cannot be solved by dividing by a "variable" (a letter) coefficient if you do not know that the "variable" is either always positive or always negative.



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