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Graphing Slope, Undefined Slope, Flat Slope Measuring Mass/Weight Interval Notation Area of a Circle Weighted Mean Area of Isosceles Triangle Factoring QuadraticsThis page looks at demonstrating some examples of solving linear inequalities in Math.

The __ Math Inequalities Introduction__ page
introduced the concept of inequalities with examples involving numbers.

But generally most inequalities encountered in Math, will involve variables, just like with equations.

A fairly basic equation involving the variable

Solving this equation means finding out which value of * x*,
added to

Recalling the properties of equality shown on the

But this equation could also be presented differently, as an inequality.

This can also be solved also.

Though in solving this time, we would be finding out what values of * x*,
added to

We can just proceed the same way as before, with subtraction on both sides.

The solutions to

Any

In fact it's the case that inequalities will often have a large range of possible solutions, rather
than just one.

So as shown, you can deal with and manipulate inequalities just like equations.

But care does need to be taken with some operations when looking to solve inequalities.

When both sides of the inequalities above were divided by **2**, each time it was only the left
and right hand sides that changed and not the symbols.

But if we look at two examples below.

dividing both sides by

However, when dividing both sides by  -

The inequality symbols in the middle need to be changed and switched around.

Why? Consider an example with just numbers.

But if we divide both sides by

So in such situations, we switch around the inequality symbol to get

This process is also the same case with multiplication.

Looking at

Both sides multiplied by the positive number

which is correct.

But both sides multiplied by the negative number

which is NOT correct.

So again, the inequality symbol is changed in order for the inequality to be correct, so that we get

Thus multiplication and division involving negative numbers is something to pay extra attention to,
when solving linear inequalities.

Knowing all this, we can list the general properties of inequality, similar to the properties of
equality.

Properties of Inequality

Assuming **a**, **b** and **c** are real numbers.

If

If

If

If

If

If

The concepts from this page for solving linear inequalities can be seen in action with examples
featured on the __ Solving Linear Inequalities__
page.

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