Solving+rational+inequalities

Date: Wednesday, Oct. 29/11
 * Solving rational inequalities**

Steps to solve a rational inequality: 1. Put the expression in the form R(x) < 0 (<,>,=, etc) 2. Determine what values of x make the expression true. Use a chart, sketch, numberline, etc to determine which intervals are positive [for R(x)>0] and negative [for R(x)<0]. For R(x)<=0 or R(x)<=0, inxclude x signs for the x-intercepts, **but NOT the asymptote or hole values.**

Ei. Solve 0 <= (x-1)(x+1) / (x+3)(x-3) i) This equation is already in the form of R(x) =>0, but you may have to do that first. ii) You do not //have// to factor the numerator and denominator, but it makes your life easier when solving for x-intercepts, and asymptotes. Split the function into intervals where the function changes from (+) to (-) This occurs at x-intercepts and **possibly** at asymptotes. x-intercepts => +1, -1 (Where numerator = 0) asymptotes => -3, +3 (Where denominator = 0) By either using a chart or a graph, determine where R(x) fits your original expression. In this case where R(x) => 0


 * < **Factor** ||= **x<-3** ||= **-33** ||
 * < (x-1) ||= - ||= - ||= - ||= + ||= + ||
 * < (x+1) ||= - ||= - ||= + ||= + ||= + ||
 * < (x+3) ||= - ||= + ||= + ||= + ||= + ||
 * < (x-3) ||= - ||= - ||= - ||= - ||= + ||
 * < **R(x)** ||= **+** ||= **-** ||= **+** ||= **-** ||= **+** ||



R(x) > 0 in the intervals x<-3, -13. We cannot include the = sign for all of these intervals as x **cannot = -3 or +3**, thus do not the values. Thus, for 0 <= (x-1)(x+1) / (x+3)(x-3),