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## Solving Linear Inequalities• An inequality is an equation with ≤,<,≥ or > instead of
an equal sign
i.e. a < b and k is any real, =>a + k < b + k • Multiplying both sides by a POSITIVE constant won’t change the inequality i.e. a < b, k > 0=>ak < bk • Multiplying both sides by a NEGATIVE constant WILL change the direction of the inequality i.e. a < b, k < 0=>ak > bk • Taking the reciprocal WILL change the direction of the inequality i.e.
– interval notation uses parenthesis, “(” and brackets, “]” – set notation uses inequality symbols, “>” – graphing uses the number line with open and closed circles connected with lines or shading • An interval can be open (meaning you don’t actually “contain” the endpoint). The open end is described with “)” “>” and/or open circles • Or an interval can be closed (meaning you do contain the endpoint). The closed end is described with “]” “ ≥ ” and/or closed circles • Example. To describe the values for x greater than 5 (5,∞) x > 5
negative is a positive. i.e. to solve for (x − 3)(x + 2) < 0 , you can’t just take when both are negative. You have to do what is called a “sign diagram” This can be done in one of two ways… 1) Look at the sign of x – 3 and x + 2 separately on a chart You can see that (x − 3)(x + 2) < 0 when –2 < x < 3 2) Still do a chart, but just “test” values to the left
and right of your “zeros” • Don’t ever just divide by x if you’re trying to solve an
inequality for x. Not only are you neglecting • When solving inequalities that contain a squared term,
you can’t just take the square root as
• Example. Solve (x +1)(x + 2) > x(x +1) • Example. Solve −3 <1− 2x ≤ 3 • Example. Solve 5x +11≤ −4 or 5x +11≥ 4
• Example. Solve |