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1

Solved example of binomial theorem

$4\left(x-1\right)^2\geq \left(2x-3\right)^2-7$
2

Grouping terms

$4\left(x-1\right)^2-\left(2x-3\right)^2\geq -7$

$4\left(x^2+2-1x+{\left(-1\right)}^2\right)-\left(2x-3\right)^2\geq -7$

Multiply $2$ times $-1$

$4\left(x^2-2x+{\left(-1\right)}^2\right)-\left(2x-3\right)^2\geq -7$

Calculate the power ${\left(-1\right)}^2$

$4\left(x^2-2x+1\right)-\left(2x-3\right)^2\geq -7$
3

Expand $\left(x-1\right)^2$

$4\left(x^2-2x+1\right)-\left(2x-3\right)^2\geq -7$
4

Solve the product $4\left(x^2-2x+1\right)$

$4x^2+4\left(-2x+1\right)-\left(2x-3\right)^2\geq -7$

$4x^2+4\cdot -2x+4\cdot 1-\left(2x-3\right)^2\geq -7$

Multiply $4$ times $-2$

$4x^2-8x+4\cdot 1-\left(2x-3\right)^2\geq -7$

Any expression multiplied by $1$ is equal to itself

$4x^2-8x+4-\left(2x-3\right)^2\geq -7$
5

Solve the product $4\left(-2x+1\right)$

$4x^2-8x+4-\left(2x-3\right)^2\geq -7$

$4\left(x^2+2-1x+{\left(-1\right)}^2\right)-\left(2x-3\right)^2\geq -7$

Multiply $2$ times $-1$

$4\left(x^2-2x+{\left(-1\right)}^2\right)-\left(2x-3\right)^2\geq -7$

Calculate the power ${\left(-1\right)}^2$

$4\left(x^2-2x+1\right)-\left(2x-3\right)^2\geq -7$

$4x^2-8x+4-\left(\left(2x\right)^2+2\cdot 2\cdot -3x+{\left(-3\right)}^2\right)\geq -7$

Multiply $2$ times $2$

$4x^2-8x+4-\left(\left(2x\right)^2+4\cdot -3x+{\left(-3\right)}^2\right)\geq -7$

Calculate the power ${\left(-3\right)}^2$

$4x^2-8x+4-\left(\left(2x\right)^2+4\cdot -3x+9\right)\geq -7$

Multiply $4$ times $-3$

$4x^2-8x+4-\left(\left(2x\right)^2-12x+9\right)\geq -7$

The power of a product is equal to the product of it's factors raised to the same power

$4x^2-8x+4-\left(4x^2-12x+9\right)\geq -7$
6

Expand $\left(2x-3\right)^2$

$4x^2-8x+4-\left(4x^2-12x+9\right)\geq -7$

$4x^2+4\cdot -2x+4\cdot 1-\left(2x-3\right)^2\geq -7$

Multiply $4$ times $-2$

$4x^2-8x+4\cdot 1-\left(2x-3\right)^2\geq -7$

Any expression multiplied by $1$ is equal to itself

$4x^2-8x+4-\left(2x-3\right)^2\geq -7$
7

Solve the product $-\left(4x^2-12x+9\right)$

$4x^2-8x+4-4x^2-\left(-12x+9\right)\geq -7$

$4x^2+4\cdot -2x+4\cdot 1-\left(2x-3\right)^2\geq -7$

Multiply $4$ times $-2$

$4x^2-8x+4\cdot 1-\left(2x-3\right)^2\geq -7$

Any expression multiplied by $1$ is equal to itself

$4x^2-8x+4-\left(2x-3\right)^2\geq -7$

$4x^2-8x+4-4x^2+12x-9\geq -7$

Subtract the values $4$ and $-9$

$-5+4x^2-8x-4x^2+12x\geq -7$
8

Solve the product $-\left(-12x+9\right)$

$-5+4x^2-8x-4x^2+12x\geq -7$
9

Adding $4x^2$ and $-4x^2$

$-5-8x+12x\geq -7$
10

Adding $-8x$ and $12x$

$4x-5\geq -7$

$4x\geq -7+5$

Subtract the values $5$ and $-7$

$4x\geq -2$
11

Moving the term $-5$ to the other side of the inequation with opposite sign

$4x\geq -2$

$x\geq \frac{-2}{4}$

Divide $-2$ by $4$

$x\geq -\frac{1}{2}$
12

Divide both sides of the inequation by $4$

$x\geq -\frac{1}{2}$

Final Answer

$x\geq -\frac{1}{2}$

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