Algebra Examples

$2.25 y^{2} - 3 y + 1 < 0$

Step 1

Convert the inequality to an equation.

$2.25 y^{2} - 3 y + 1 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $0.25$ out of $2.25 y^{2} - 3 y + 1$ .

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Step 2.1.1

Factor $0.25$ out of $2.25 y^{2}$ .

$0.25 (9 y^{2}) - 3 y + 1 = 0$

Step 2.1.2

Factor $0.25$ out of $- 3 y$ .

$0.25 (9 y^{2}) + 0.25 (- 12 y) + 1 = 0$

Step 2.1.3

Factor $0.25$ out of $1$ .

$0.25 (9 y^{2}) + 0.25 (- 12 y) + 0.25 (4) = 0$

Step 2.1.4

Factor $0.25$ out of $0.25 (9 y^{2}) + 0.25 (- 12 y)$ .

$0.25 (9 y^{2} - 12 y) + 0.25 (4) = 0$

Step 2.1.5

Factor $0.25$ out of $0.25 (9 y^{2} - 12 y) + 0.25 (4)$ .

$0.25 (9 y^{2} - 12 y + 4) = 0$

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

Rewrite $9 y^{2}$ as ${(3 y)}^{2}$ .

$0.25 ({(3 y)}^{2} - 12 y + 4) = 0$

Step 2.2.2

Rewrite $4$ as $2^{2}$ .

$0.25 ({(3 y)}^{2} - 12 y + 2^{2}) = 0$

Step 2.2.3

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

$12 y = 2 \cdot (3 y) \cdot 2$

Step 2.2.4

Rewrite the polynomial.

$0.25 ({(3 y)}^{2} - 2 \cdot (3 y) \cdot 2 + 2^{2}) = 0$

Step 2.2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 3 y$ and $b = 2$ .

$0.25 {(3 y - 2)}^{2} = 0$

Step 3

Divide each term in $0.25 {(3 y - 2)}^{2} = 0$ by $0.25$ and simplify.

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Step 3.1

Divide each term in $0.25 {(3 y - 2)}^{2} = 0$ by $0.25$ .

$\frac{0.25 {(3 y - 2)}^{2}}{0.25} = \frac{0}{0.25}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $0.25$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{0.25 {(3 y - 2)}^{2}}{0.25} = \frac{0}{0.25}$

Step 3.2.1.2

Divide ${(3 y - 2)}^{2}$ by $1$ .

${(3 y - 2)}^{2} = \frac{0}{0.25}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $0.25$ .

${(3 y - 2)}^{2} = 0$

Step 4

Set the $3 y - 2$ equal to $0$ .

$3 y - 2 = 0$

Step 5

Solve for $y$ .

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Step 5.1

Add $2$ to both sides of the equation.

$3 y = 2$

Step 5.2

Divide each term in $3 y = 2$ by $3$ and simplify.

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Step 5.2.1

Divide each term in $3 y = 2$ by $3$ .

$\frac{3 y}{3} = \frac{2}{3}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{2}{3}$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{2}{3}$

Step 6

Use each root to create test intervals.

$y < \frac{2}{3}$

$y > \frac{2}{3}$

Step 7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 7.1

Test a value on the interval $y < \frac{2}{3}$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $y < \frac{2}{3}$ and see if this value makes the original inequality true.

$y = 0$

Step 7.1.2

Replace $y$ with $0$ in the original inequality.

$2.25 {(0)}^{2} - 3 \cdot 0 + 1 < 0$

Step 7.1.3

The left side $1$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 7.2

Test a value on the interval $y > \frac{2}{3}$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $y > \frac{2}{3}$ and see if this value makes the original inequality true.

$y = 3$

Step 7.2.2

Replace $y$ with $3$ in the original inequality.

$2.25 {(3)}^{2} - 3 \cdot 3 + 1 < 0$

Step 7.2.3

The left side $12.25$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 7.3

Compare the intervals to determine which ones satisfy the original inequality.

$y < \frac{2}{3}$ False

$y > \frac{2}{3}$ False

$y < \frac{2}{3}$ False

$y > \frac{2}{3}$ False

Step 8

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution