Solve the Inequality for x 121x^2-22x+1<0

$121 x^{2} - 22 x + 1 < 0$

Step 1

Convert the inequality to an equation.

$121 x^{2} - 22 x + 1 = 0$

Step 2

Factor using the perfect square rule.

Rewrite $121 x^{2}$ as ${(11 x)}^{2}$ .

${(11 x)}^{2} - 22 x + 1 = 0$

Rewrite $1$ as $1^{2}$ .

${(11 x)}^{2} - 22 x + 1^{2} = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$22 x = 2 \cdot (11 x) \cdot 1$

Rewrite the polynomial.

${(11 x)}^{2} - 2 \cdot (11 x) \cdot 1 + 1^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 11 x$ and $b = 1$ .

${(11 x - 1)}^{2} = 0$

Step 3

Set the $11 x - 1$ equal to $0$ .

$11 x - 1 = 0$

Step 4

Solve for $x$ .

Add $1$ to both sides of the equation.

$11 x = 1$

Divide each term in $11 x = 1$ by $11$ and simplify.

Divide each term in $11 x = 1$ by $11$ .

$\frac{11 x}{11} = \frac{1}{11}$

Simplify the left side.

Cancel the common factor of $11$ .

Cancel the common factor.

$\frac{11 x}{11} = \frac{1}{11}$

Divide $x$ by $1$ .

$x = \frac{1}{11}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{11}$

Decimal Form:

$x = 0.\overline{09}$

Step 6