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web-inequalities-john - Solving inequalities Inequalities...

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Solving inequalities Inequalities are mathematical expressions involving the symbols > , < , and . To ‘solve’ an inequality means to find a range, or ranges, of values that an unknown x can take and still satisfy the inequality. In this unit inequalities are solved by using algebra and by using graphs. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: solve simple inequalities using algebra solve simple inequalities by drawing graphs solve inequalities in which there is a modulus symbol solve quadratic inequalities Contents 1. Introduction 2 2. Manipulation of inequalities 2 3. Solving some simple inequalities 3 4. Inequalities used with a modulus symbol 5 5. Using graphs to solve inequalities 7 6. Quadratic inequalities 8 1 c math centre August 7, 2003

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1. Introduction The expression 5 x 4 > 2 x + 3 looks like an equation but with the equals sign replaced by an arrowhead. It is an example of an inequality . This denotes that the part on the left, 5 x 4, is greater than the part on the right, 2 x + 3. We will be interested in finding the values of x for which the inequality is true. We use four symbols to denote inequalities: Key Point > is greater than is greater than or equal to < is less than is less than or equal to Notice that the arrowhead always points to the smaller expression. 2. Manipulation of inequalities Inequalities can be manipulated like equations and follow very similar rules, but there is one important exception. If you add the same number to both sides of an inequality, the inequality remains true. If you subtract the same number from both sides of the inequality, the inequality remains true. If you multiply or divide both sides of an inequality by the same positive number, the inequality remains true. But if you multiply or divide both sides of an inequality by a negative number, the inequality is no longer true. In fact, the inequality becomes reversed. This is quite easy to see because we can write that 4 > 2. But if we multiply both sides of this inequality by 1, we get 4 > 2, which is not true. We have to reverse the inequality, giving 4 < 2 in order for it to be true. This leads to diﬃculties when dealing with variables, because a variable can be either positive or negative. Consider the inequality x 2 > x It looks as though we might be able to divide both sides by x to give x > 1 c math centre August 7, 2003 2
But, in fact, we cannot do this. The two inequalities x 2 > x and x > 1 are not the same. This is because in the inequality x > 1, x is clearly greater than 1. But in the inequality x 2 > x we have to take into account the possibility that x is negative, since if x is negative, x 2 (which must be positive or zero) is always greater than x . In fact the complete solution of this inequality is x > 1 or x < 0. The second part of the solution must be true since if x

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