Solve for x: 3.1 x2-2x-3>0, 3.2 x2+x-6<0, 3.3 x2-2x≥0, 3.4 x2+x-12≤0, 3.5 2-x-x2>0, 3.6 10+x≤2x2, 3.7 x2>4

3.1 x^2 - 2x - 3 > 0 Step 1: Find the roots of the quadratic equation x^2 - 2x - 3 = 0. (x-3)(x+1) = 0 x = 3 or x = -1 Step 2: Sketch the parabola y = x^2 - 2x - 3. Since the leading coefficient is positive, the parabola opens upwards. The inequality x^2 - 2x - 3 > 0 means we are looking for the values of x where the parabola is above the x-axis. Solution: x < -1 or x > 3 3.2 x^2 + x - 6 < 0 Step 1: Find the roots of the quadratic equation x^2 + x - 6 = 0. (x+3)(x-2) = 0 x = -3 or x = 2 Step 2: Sketch the parabola y = x^2 + x - 6. Since the leading coefficient is positive, the parabola opens upwards. The inequality x^2 + x - 6 < 0 means we are looking for the values of x where the parabola is below the x-axis. Solution: -3 < x < 2 3.3 x^2 - 2x 0 Step 1: Find the roots of the quadratic equation x^2 - 2x = 0. x(x-2) = 0 x = 0 or x = 2 Step 2: Sketch the parabola y = x^2 - 2x. Since the leading coefficient is positive, the parabola opens upwards. The inequality x^2 - 2x 0 means we are looking for the values of x where the parabola is above or on the x-axis. Solution: x 0 or x 2 3.4 x^2 + x - 12 0 Step 1: Find the roots of the quadratic equation x^2 + x - 12 = 0. (x+4)(x-3) = 0 x = -4 or x = 3 Step 2: Sketch the parabola y = x^2 + x - 12. Since the leading coefficient is positive, the parabola opens upwards. The inequality x^2 + x - 12 0 means we are looking for the values of x where the parabola is below or on the x-axis. Solution: -4 x 3 3.5 2 - x - x^2 > 0 Step 1: Rewrite the inequality by multiplying by -1 and reversing the sign. -x^2 - x + 2 > 0 x^2 + x - 2 < 0 Step 2: Find the roots of the quadratic equation x^2 + x - 2 = 0. (x+2)(x-1) = 0 x = -2 or x = 1 Step 3: Sketch the parabola y = x^2 + x - 2. Since the leading coefficient is positive, the parabola opens upwards. The inequality x^2 + x - 2 < 0 means we are looking for the values of x where the parabola is below the x-axis. Solution: -2 < x < 1 3.6 10 + x 2x^2 Step 1: Rearrange the inequality to have 0 on one side. 0 2x^2 - x - 10 2x^2 - x - 10 0 Step 2: Find the roots of the quadratic equation 2x^2 - x - 10 = 0 using the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. x = -(-1) ± sqrt((-1)^2 - 4(2)(-10))2(2) x = 1 ± sqrt(1 + 80)4 x = 1 ± sqrt(81)4 x = (1 ± 9)/(4) x_1 = (1 + 9)/(4) = (10)/(4) = (5)/(2) x_2 = (1 - 9)/(4) = (-8)/(4) = -2 Step 3: Sketch the parabola y = 2x^2 - x - 10. Since the leading coefficient is positive, the parabola opens upwards. The inequality 2x^2 - x - 10 0 means we are looking for the values of x where the parabola is above or on the x-axis. Solution: x -2 or x (5)/(2) 3.7 x^2 > 4 Step 1: Rearrange the inequality to have 0 on one side. x^2 - 4 > 0 Step 2: Find the roots of the quadratic equation x^2 - 4 = 0. (x-2)(x+2) = 0 x = 2 or x = -2 Step 3: Sketch the parabola y = x^2 - 4. Since the leading coefficient is positive, the parabola opens upwards. The inequality x^2 - 4 > 0 means we are looking for the values of x where the parabola is above the x-axis. Solution: x < -2 or x > 2