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**Unformatted text preview: **MA100, Fall 2010, PART 1: Precalculus [4] Linear Equations and Inequalities. Absolute Value 1. Linear equations Definition A mathematical expression of the form ax + b where a,b ∈ R is called a linear expression . For instance, 3 x- 7 is a linear expression with a = 3 and b =- 7 . Solving ax + b = 0 , where a,b ∈ R Let a,b ∈ R , and consider the equation ax + b = 0 . We have the following: ax + b = 0 = ⇒ ax =- b Now: (A) if a 6 = 0 then the equation has an unique solution (i.e. one solution only) which is given by x =- b a ; (B) if a = 0 then there are two possibilities: either b = 0 or b 6 = 0 . i) If b = 0 then any x ∈ R is a solution. This is because in this case our equation ax + b = 0 becomes 0 · x = 0 , that is 0 = 0. In other words, we obtain have 0 = 0 no matter what x is. But 0 = 0 is always true. In this case, we say that the equation ax + b = 0 infinitely many solutions. ii) If b 6 = 0 then there is no solution. This is because in this case the equation ax + b = 0 becomes 0 · x = b , or 0 = b. Since b 6 = 0, we cannot find any x that would solve the equation. Example: Solve (23- 23) x = 7 . Answer: The equation reads × x = 7 = ⇒ 0 = 7 = ⇒ there is no solution (or, equivalently, the solution set is void) Example: Find m ∈ R and n ∈ R so that the equation for x, where x ∈ R , (3 m- 2) x = (2 n- 6) has infinitely many solutions. 1 Answer: An equation of the form ax =- b has infinitely many solutions if a = b = 0 . In our case, since a = 3 m- 2 and b = 2 n- 6, this means that we have to impose the conditions: 3 m- 2 = 0 and 2 n- 6 = 0 Solving the above for m we have 3 m- 2 = 0 = ⇒ 3 m = 2 = ⇒ m = 2 3 and solving for n we obtain 2 n- 6 = 0 = ⇒ 2 n = 6 = ⇒ n = 6 2 = ⇒ n = 3 . So for m = 2 3 and n = 3, the equation (3 m- 2) x = (2 n- 6) becomes 0 · x = 0 and thus it has infinitely many solutions (any x ∈ R is a solution)....

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