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Inequalities

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While in an equation the "=" sign indicates that the sides are identical, inequalities are used when they aren't. Different inequalities represent different relationships between the two sides. The definitions of the different symbols that can be used in inequalities are given in the diagram.

Inequalities can be represented on numbre lines, where a hollow dot represents < or >, and a solid dot represents ≤ or ≥. To draw a number line, draw a circle over the relevant number, either hollow or solid as appropriate, then draw an arrow in the direction that makes the inequality true, ie. to the left if it is less than or to the right if it is more than.

Linear inequalities can be solved just as linear equations using inverse operations, with the only difference being that the inequality sign is kept throughout rather than the "=".

To solve quadratic inequalities, rearrange them so that one side is 0. Then solve the corresponding quadratic equation. Sketch the graph of the quadratic equation using the roots and deduce the range of values the variable can take. When the quadratic expression is greater than 0, the wanted region is above the x-axis, and when the quadratic expression is less than 0, the wanted region is below the x-axis.

When solving inequalities, if it is required to multiply or divide by a negative number, the direction of the inequality sign must be reversed. When there is an expression in terms of the variable that is being solved in the denominator, multiply every term by the square of the expression, so that there is no need to reverse the inequality sign, as this will guarantee that the number that the terms are being multiplied by is positive.