Now we have the inequality: -12x^2 + 39x - 34 < 13x - 26x^2
Let's simplify further by moving all terms to one side of the inequality: -12x^2 + 39x - 34 - 13x + 26x^2 < 0 14x^2 + 26x - 34 < 0
Now we need to solve this quadratic inequality. Let's first find the roots of the quadratic equation 14x^2 + 26x - 34 = 0 by using the quadratic formula:
x = [-26 ± √(26^2 - 4(14)(-34))] / 2(14) x = [-26 ± √(676 + 1904)] / 28 x = (-26 ± √2580) / 28 x = (-26 ± 50.79) / 28
Now we have two possible solutions: x1 = (-26 + 50.79) / 28 ≈ 0.91 x2 = (-26 - 50.79) / 28 ≈ -2.60
So, the solution to the inequality is: -2.6 < x < 0.91
Let's first simplify both sides of the inequality:
Left side:
(4x - 5)(6 - 3x) - 4
= 24x - 12x^2 - 30 + 15x - 4
= -12x^2 + 39x - 34
Right side:
(1 - 2x)(7x + 6x)
= (1 - 2x)(13x)
= 13x - 26x^2
Now we have the inequality:
-12x^2 + 39x - 34 < 13x - 26x^2
Let's simplify further by moving all terms to one side of the inequality:
-12x^2 + 39x - 34 - 13x + 26x^2 < 0
14x^2 + 26x - 34 < 0
Now we need to solve this quadratic inequality. Let's first find the roots of the quadratic equation 14x^2 + 26x - 34 = 0 by using the quadratic formula:
x = [-26 ± √(26^2 - 4(14)(-34))] / 2(14)
x = [-26 ± √(676 + 1904)] / 28
x = (-26 ± √2580) / 28
x = (-26 ± 50.79) / 28
Now we have two possible solutions:
x1 = (-26 + 50.79) / 28 ≈ 0.91
x2 = (-26 - 50.79) / 28 ≈ -2.60
So, the solution to the inequality is: -2.6 < x < 0.91