02Iinequations

= Solving Inequations = toc Recall that an equation has an equals sign ( = )

An ** inequation ** has an inequality sign ( <, > , __<__ , __>__)

Note: Many text books and web sites (including this one, sometimes) show:
 * the " less than or equal to " sign as __<__
 * the " greater than or equal to " sign as __>__

This is due to limitations in the character set available. The second line should be sloping and not horizontal.

math . \qquad \text{You should always write } \leqslant \text{ and not } \underline{<} math

math . \qquad \text{Similarly, you should always write } \geqslant \text{ and not } \underline{>} math

Inequalities
The solution to an inequation will be something like **x > 1** which means that all values of x larger than 1 (including fractions/decimals) are solutions to the inequation.

On the number line, x > 1 can be shown like this:




 * The open circle above the 1 indicates that 1 is not included but all values above 1 are included.

On the number line, x ≤ 2 can be shown like this:
 * The closed (or filled in) circle indicates that the 2 is included.

Solving Inequations
For most situations, follow exactly the same rules as for solving equations, except that you are writing an inequality sign instead of an equals sign.

** Example 1 **
math \text{Solve : } 2x + 3 > 7 math

__**Solution:**__

math \\ . \qquad 2x+3 > 7 \qquad \{ -3 \} \\. \\ . \qquad 2x > 4 \qquad \qquad \{ \div 2 \} \\. \\ . \qquad x > 2 math

Rearranging Inequations
If you end up with the variable on the right side, it is important to rewrite it with the variable on the left. As you do so, __reverse__ the direction of the inequality sign.

math \text{This is because } 6 \leqslant x \text{ is equaivalent to } x \geqslant 6 math



** Example 2 **
math \text{Solve : } 3x + 6 \leqslant 4x math

__**Solution:**__

math \\ . \qquad 3x+6 \leqslant 4x \qquad \{ -3x \} \\. \\ . \qquad 6 \leqslant x \qquad \qquad \qquad \{ \textit{Reverse} \} \\. \\ . \qquad x \geqslant 6 math

Multiplying or Dividing by a Negative
There is __one__ rule you need to apply when solving inequations (compared to solving equations).

When multiplying or dividing by a negative number, __reverse__ the direction of the inequality sign.

So, for example math . \qquad -x < -2 \; \textit{ is equivalent to } \; x > 2 math


 * if –x = –2, then x = 2
 * if –x = –3, then x = 3
 * etc
 * so any values of –x less than –2 are equivalent to the values of x being greater than 2

** Example 3 **
math \text{Solve : } -2x + 1 < -3 math

__**Solution:**__

math \\ . \qquad -2x+1<-3 \qquad \{ -1 \} \\. \\ . \qquad -2x < -4 \qquad \quad \; \{ \div -2 \} \; \textit{ so reverse sign} \\. \\ . \qquad x > 2 math

For another site that explains this idea, go here: MathsIsFun Go to top of page flat

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