{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Ch4_LinearIneq

# Ch4_LinearIneq - MA100 Fall 2010 PART 1 Precalculus[4...

This preview shows pages 1–3. Sign up to view the full content.

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 0 × 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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).
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}