Solve (1/2-x)^2-(x+1)^2<-[1-(2x+1/6)]+1+2/3

$$( \frac { 1 } { 2 } - x ) ^ { 2 } - ( x + 1 ) ^ { 2 } < - [ 1 - ( \frac { 2 x + 1 } { 6 } ) ] + \frac { 1 + 2 } { 3 }$$

Solve for x

$$x>-\frac{11}{40}$$

Solution Steps

Multiply both sides of the equation by 3. Since 3 is positive, the inequality direction remains the same.

$$3\left(\frac{1}{2}-x\right)^{2}-3\left(x+1\right)^{2}<-3\left(1-\frac{2x+1}{6}\right)+1+2$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{1}{2}-x\right)^{2}.

$$3\left(\frac{1}{4}-x+x^{2}\right)-3\left(x+1\right)^{2}<-3\left(1-\frac{2x+1}{6}\right)+1+2$$

Use the distributive property to multiply 3 by \frac{1}{4}-x+x^{2}.

$$\frac{3}{4}-3x+3x^{2}-3\left(x+1\right)^{2}<-3\left(1-\frac{2x+1}{6}\right)+1+2$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$\frac{3}{4}-3x+3x^{2}-3\left(x^{2}+2x+1\right)<-3\left(1-\frac{2x+1}{6}\right)+1+2$$

Use the distributive property to multiply -3 by x^{2}+2x+1.

$$\frac{3}{4}-3x+3x^{2}-3x^{2}-6x-3<-3\left(1-\frac{2x+1}{6}\right)+1+2$$

Combine 3x^{2} and -3x^{2} to get 0.

$$\frac{3}{4}-3x-6x-3<-3\left(1-\frac{2x+1}{6}\right)+1+2$$

Combine -3x and -6x to get -9x.

$$\frac{3}{4}-9x-3<-3\left(1-\frac{2x+1}{6}\right)+1+2$$

Subtract 3 from \frac{3}{4} to get -\frac{9}{4}.

$$-\frac{9}{4}-9x<-3\left(1-\frac{2x+1}{6}\right)+1+2$$

Use the distributive property to multiply -3 by 1-\frac{2x+1}{6}.

$$-\frac{9}{4}-9x<-3-3\left(-\frac{2x+1}{6}\right)+1+2$$

Multiply -3 and -1 to get 3.

$$-\frac{9}{4}-9x<-3+3\times \frac{2x+1}{6}+1+2$$

Cancel out 6, the greatest common factor in 3 and 6.

$$-\frac{9}{4}-9x<-3+\frac{2x+1}{2}+1+2$$

Add -3 and 1 to get -2.

$$-\frac{9}{4}-9x<-2+\frac{2x+1}{2}+2$$

Add -2 and 2 to get 0.

$$-\frac{9}{4}-9x<\frac{2x+1}{2}$$

Divide each term of 2x+1 by 2 to get x+\frac{1}{2}.

$$-\frac{9}{4}-9x<x+\frac{1}{2}$$

Subtract x from both sides.

$$-\frac{9}{4}-9x-x<\frac{1}{2}$$

Combine -9x and -x to get -10x.

$$-\frac{9}{4}-10x<\frac{1}{2}$$

Add \frac{9}{4} to both sides.

$$-10x<\frac{1}{2}+\frac{9}{4}$$

Add \frac{1}{2} and \frac{9}{4} to get \frac{11}{4}.

$$-10x<\frac{11}{4}$$

Divide both sides by -10. Since -10 is negative, the inequality direction is changed.

$$x>\frac{\frac{11}{4}}{-10}$$

Express \frac{\frac{11}{4}}{-10} as a single fraction.

$$x>\frac{11}{4\left(-10\right)}$$

Multiply 4 and -10 to get -40.

$$x>\frac{11}{-40}$$

Fraction \frac{11}{-40} can be rewritten as -\frac{11}{40} by extracting the negative sign.

$$x>-\frac{11}{40}$$