find the smallest natural number of 4x-3/2 -x greater than 2/3 (x+1)

To find the smallest natural number that satisfies the inequality (4x - 3)/(2 - x) > (2/3)(x + 1), we can follow these steps: Multiply both sides of the inequality by (2 - x) to eliminate the denominator: (2 - x) * (4x - 3)/(2 - x) > (2/3)(x + 1) * (2 - x) Simplifying: 4x - 3 > (2/3)(x + 1) * (2 - x) Expand the right side of the inequality: 4x - 3 > (2/3)(2 - x)(x + 1) Simplifying further: 4x - 3 > (2/3)(2x - x^2 + 2 - x) Distribute the (2/3) term: 4x - 3 > (4/3)x - (2/3)x^2 + (4/3) - (2/3)x Combine like terms: 4x - 3 > (4/3)x - (2/3)x^2 + (4/3) - (2/3)x Rearranging the terms: (2/3)x^2 + (2/3)x - 7/3 > 0 Multiply both sides of the inequality by 3 to get rid of the fractions: 2x^2 + 2x - 7 > 0 Now we need to solve this quadratic inequality. To determine the smallest natural number that satisfies the inequality, we can factor the quadratic expression or use the quadratic formula. Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a) For the quadratic equation 2x^2 + 2x - 7 > 0, we have: a = 2, b = 2, c = -7 x = (-2 ± √(2^2 - 4(2)(-7))) / (2(2)) x = (-2 ± √(4 + 56)) / 4 x = (-2 ± √60) / 4 x = (-2 ± 2√15) / 4 x = (-1 ± √15) / 2 The two possible solutions are: x = (-1 + √15) / 2 x = (-1 - √15) / 2 To find the smallest natural number that satisfies the inequality, we need to evaluate these solutions. Evaluating (-1 + √15) / 2 ≈ 0.8728, which is not a natural number. Evaluating (-1 - √15) / 2 ≈ -1.8728, which is not a natural number. Therefore, there is no natural number that satisfies the inequality (4x - 3)/(2 - x) > (2/3)(x + 1).