Solve 4(x-3)^2-(2x-5)^2geq2

$$4 ( x - 3 ) ^ { 2 } - ( 2 x - 5 ) ^ { 2 } \geq 2$$

Solve for x

$$x\leq \frac{9}{4}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$4\left(x^{2}-6x+9\right)-\left(2x-5\right)^{2}\geq 2$$

Use the distributive property to multiply 4 by x^{2}-6x+9.

$$4x^{2}-24x+36-\left(2x-5\right)^{2}\geq 2$$

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

$$4x^{2}-24x+36-\left(4x^{2}-20x+25\right)\geq 2$$

To find the opposite of 4x^{2}-20x+25, find the opposite of each term.

$$4x^{2}-24x+36-4x^{2}+20x-25\geq 2$$

Combine 4x^{2} and -4x^{2} to get 0.

$$-24x+36+20x-25\geq 2$$

Combine -24x and 20x to get -4x.

$$-4x+36-25\geq 2$$

Subtract 25 from 36 to get 11.

$$-4x+11\geq 2$$

Subtract 11 from both sides.

$$-4x\geq 2-11$$

Subtract 11 from 2 to get -9.

$$-4x\geq -9$$

Divide both sides by -4. Since -4 is negative, the inequality direction is changed.

$$x\leq \frac{-9}{-4}$$

Fraction \frac{-9}{-4} can be simplified to \frac{9}{4} by removing the negative sign from both the numerator and the denominator.

$$x\leq \frac{9}{4}$$