Solve (x-1)^2geq(5-x)^2

$$( x - 1 ) ^ { 2 } \geq ( 5 - x ) ^ { 2 }$$

Solve for x

$$x\geq 3$$

Solution Steps

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-x\right)^{2}.

$$x^{2}-2x+1\geq 25-10x+x^{2}$$

Add 10x to both sides.

$$x^{2}-2x+1+10x\geq 25+x^{2}$$

Combine -2x and 10x to get 8x.

$$x^{2}+8x+1\geq 25+x^{2}$$

Subtract x^{2} from both sides.

$$x^{2}+8x+1-x^{2}\geq 25$$

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

$$8x+1\geq 25$$

Subtract 1 from both sides.

$$8x\geq 25-1$$

Subtract 1 from 25 to get 24.

$$8x\geq 24$$

Divide both sides by 8. Since 8 is positive, the inequality direction remains the same.

$$x\geq \frac{24}{8}$$

Divide 24 by 8 to get 3.

$$x\geq 3$$