This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Rewrite the inequality by moving all terms to one side. The given inequality is: (3)/(5x-2) (2)/(3x+1) Subtract (2)/(3x+1) from both sides: (3)/(5x-2) - (2)/(3x+1) 0 Step 2: Find a common denominator and combine the fractions. The common denominator is (5x-2)(3x+1). (3(3x+1) - 2(5x-2))/((5x-2)(3x+1)) 0 Expand the numerator: (9x+3 - 10x+4)/((5x-2)(3x+1)) 0 Simplify the numerator: (-x+7)/((5x-2)(3x+1)) 0 Step 3: Identify the critical points. Critical points are the values of x that make the numerator zero or the denominator zero. For the numerator: -x+7 = 0 x = 7. For the denominator: 5x-2 = 0 5x = 2 x = (2)/(5). 3x+1 = 0 3x = -1 x = -(1)/(3). The critical points, in increasing order, are x = -(1)/(3), x = (2)/(5), and x = 7. These points divide the number line into four intervals: (-, -(1)/(3)), (-(1)/(3), (2)/(5)), ((2)/(5), 7), and (7, ). Step 4: Test a value from each interval to determine the sign of the expression f(x) = (-x+7)/((5x-2)(3x+1)). Interval 1: x < -(1)/(3) (e.g., test x = -1) f(-1) = (-(-1)+7)/((5(-1)-2)(3(-1)+1)) = (1+7)/((-5-2)(-3+1)) = (8)/((-7)(-2)) = (8)/(14) > 0. Interval 2: -(1)/(3) < x < (2)/(5) (e.g., test x = 0) f(0) = (-0+7)/((5(0)-2)(3(0)+1)) = (7)/((-2)(1)) = (7)/(-2) < 0. Interval 3: (2)/(5) < x < 7 (e.g., test x = 1) f(1) = (-1+7)/((5(1)-2)(3(1)+1)) = (6)/((3)(4)) = (6)/(12) > 0. Interval 4: x > 7 (e.g., test x = 8) f(8) = (-8+7)/((5(8)-2)(3(8)+1)) = (-1)/((38)(25)) < 0. Step 5: Determine the solution set. We are looking for where f(x) 0. This occurs when f(x) is negative or zero. The expression is negative in the intervals (-(1)/(3), (2)/(5)) and (7, ). The expression is zero when x=7. The values x = -(1)/(3) and x = (2)/(5) make the denominator zero, so they are undefined and must be excluded (represented by open circles on the number line). Therefore, the solution set is x (-(1)/(3), (2)/(5)) [7, ). Step 6: Indicate the answer on a number line. Draw a number line, mark the critical points, and shade the regions corresponding to the solution. Use open circles for excluded points and a closed circle for included points. ` <-------------------------------------------------------------------------------------> -1/3 2/5 7 <-----o==============o----------------------------------●=============================> ` The solution is x (-(1)/(3), (2)/(5)) [7, ). 3 done, 2 left today. You're making progress.