Solve 1/3x^2+1/2-9x+4/12+5-x^2/3geqfrac{2x-1}{9}

$$\frac { 1 } { 3 } x ^ { 2 } + \frac { 1 } { 2 } - \frac { 9 x + 4 } { 12 } + \frac { 5 - x ^ { 2 } } { 3 } \geq \frac { 2 x - 1 } { 9 }$$

Solve for x

$$x\leq 2$$

Solution Steps

Multiply both sides of the equation by 36, the least common multiple of 3,2,12,9. Since 36 is positive, the inequality direction remains the same.

$$12x^{2}+18-3\left(9x+4\right)+12\left(5-x^{2}\right)\geq 4\left(2x-1\right)$$

Use the distributive property to multiply -3 by 9x+4.

$$12x^{2}+18-27x-12+12\left(5-x^{2}\right)\geq 4\left(2x-1\right)$$

Subtract 12 from 18 to get 6.

$$12x^{2}+6-27x+12\left(5-x^{2}\right)\geq 4\left(2x-1\right)$$

Use the distributive property to multiply 12 by 5-x^{2}.

$$12x^{2}+6-27x+60-12x^{2}\geq 4\left(2x-1\right)$$

Add 6 and 60 to get 66.

$$12x^{2}+66-27x-12x^{2}\geq 4\left(2x-1\right)$$

Combine 12x^{2} and -12x^{2} to get 0.

$$66-27x\geq 4\left(2x-1\right)$$

Use the distributive property to multiply 4 by 2x-1.

$$66-27x\geq 8x-4$$

Subtract 8x from both sides.

$$66-27x-8x\geq -4$$

Combine -27x and -8x to get -35x.

$$66-35x\geq -4$$

Subtract 66 from both sides.

$$-35x\geq -4-66$$

Subtract 66 from -4 to get -70.

$$-35x\geq -70$$

Divide both sides by -35. Since -35 is negative, the inequality direction is changed.

$$x\leq \frac{-70}{-35}$$

Divide -70 by -35 to get 2.

$$x\leq 2$$