Algebra Examples

$\sqrt{9 y + 19} - \sqrt{6 y - 5} > 3$

Step 1

Add $\sqrt{6 y - 5}$ to both sides of the inequality.

$\sqrt{9 y + 19} > 3 + \sqrt{6 y - 5}$

Step 2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{9 y + 19}}^{2} > {(3 + \sqrt{6 y - 5})}^{2}$

Step 3

Simplify each side of the inequality.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 y + 19}$ as ${(9 y + 19)}^{\frac{1}{2}}$ .

${({(9 y + 19)}^{\frac{1}{2}})}^{2} > {(3 + \sqrt{6 y - 5})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(9 y + 19)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(9 y + 19)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(9 y + 19)}^{\frac{1}{2} \cdot 2} > {(3 + \sqrt{6 y - 5})}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(9 y + 19)}^{\frac{1}{2} \cdot 2} > {(3 + \sqrt{6 y - 5})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(9 y + 19)}^{1} > {(3 + \sqrt{6 y - 5})}^{2}$

Step 3.2.1.2

Simplify.

$9 y + 19 > {(3 + \sqrt{6 y - 5})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(3 + \sqrt{6 y - 5})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(3 + \sqrt{6 y - 5})}^{2}$ as $(3 + \sqrt{6 y - 5}) (3 + \sqrt{6 y - 5})$ .

$9 y + 19 > (3 + \sqrt{6 y - 5}) (3 + \sqrt{6 y - 5})$

Step 3.3.1.2

Expand $(3 + \sqrt{6 y - 5}) (3 + \sqrt{6 y - 5})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$9 y + 19 > 3 (3 + \sqrt{6 y - 5}) + \sqrt{6 y - 5} (3 + \sqrt{6 y - 5})$

Step 3.3.1.2.2

Apply the distributive property.

$9 y + 19 > 3 \cdot 3 + 3 \sqrt{6 y - 5} + \sqrt{6 y - 5} (3 + \sqrt{6 y - 5})$

Step 3.3.1.2.3

Apply the distributive property.

$9 y + 19 > 3 \cdot 3 + 3 \sqrt{6 y - 5} + \sqrt{6 y - 5} \cdot 3 + \sqrt{6 y - 5} \sqrt{6 y - 5}$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $3$ by $3$ .

$9 y + 19 > 9 + 3 \sqrt{6 y - 5} + \sqrt{6 y - 5} \cdot 3 + \sqrt{6 y - 5} \sqrt{6 y - 5}$

Step 3.3.1.3.1.2

Move $3$ to the left of $\sqrt{6 y - 5}$ .

$9 y + 19 > 9 + 3 \sqrt{6 y - 5} + 3 \cdot \sqrt{6 y - 5} + \sqrt{6 y - 5} \sqrt{6 y - 5}$

Step 3.3.1.3.1.3

Multiply $\sqrt{6 y - 5} \sqrt{6 y - 5}$ .

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Step 3.3.1.3.1.3.1

Raise $\sqrt{6 y - 5}$ to the power of $1$ .

$9 y + 19 > 9 + 3 \sqrt{6 y - 5} + 3 \sqrt{6 y - 5} + {\sqrt{6 y - 5}}^{1} \sqrt{6 y - 5}$

Step 3.3.1.3.1.3.2

Raise $\sqrt{6 y - 5}$ to the power of $1$ .

$9 y + 19 > 9 + 3 \sqrt{6 y - 5} + 3 \sqrt{6 y - 5} + {\sqrt{6 y - 5}}^{1} {\sqrt{6 y - 5}}^{1}$

Step 3.3.1.3.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$9 y + 19 > 9 + 3 \sqrt{6 y - 5} + 3 \sqrt{6 y - 5} + {\sqrt{6 y - 5}}^{1 + 1}$

Step 3.3.1.3.1.3.4

Add $1$ and $1$ .

$9 y + 19 > 9 + 3 \sqrt{6 y - 5} + 3 \sqrt{6 y - 5} + {\sqrt{6 y - 5}}^{2}$

Step 3.3.1.3.1.4

Rewrite ${\sqrt{6 y - 5}}^{2}$ as $6 y - 5$ .

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Step 3.3.1.3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 y - 5}$ as ${(6 y - 5)}^{\frac{1}{2}}$ .

$9 y + 19 > 9 + 3 \sqrt{6 y - 5} + 3 \sqrt{6 y - 5} + {({(6 y - 5)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$9 y + 19 > 9 + 3 \sqrt{6 y - 5} + 3 \sqrt{6 y - 5} + {(6 y - 5)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$9 y + 19 > 9 + 3 \sqrt{6 y - 5} + 3 \sqrt{6 y - 5} + {(6 y - 5)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.4.4.1

Cancel the common factor.

$9 y + 19 > 9 + 3 \sqrt{6 y - 5} + 3 \sqrt{6 y - 5} + {(6 y - 5)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4.2

Rewrite the expression.

$9 y + 19 > 9 + 3 \sqrt{6 y - 5} + 3 \sqrt{6 y - 5} + {(6 y - 5)}^{1}$

Step 3.3.1.3.1.4.5

Simplify.

$9 y + 19 > 9 + 3 \sqrt{6 y - 5} + 3 \sqrt{6 y - 5} + 6 y - 5$

Step 3.3.1.3.2

Subtract $5$ from $9$ .

$9 y + 19 > 4 + 3 \sqrt{6 y - 5} + 3 \sqrt{6 y - 5} + 6 y$

Step 3.3.1.3.3

Add $3 \sqrt{6 y - 5}$ and $3 \sqrt{6 y - 5}$ .

$9 y + 19 > 4 + 6 \sqrt{6 y - 5} + 6 y$

Step 4

Solve for $6 \sqrt{6 y - 5}$ .

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Step 4.1

Rewrite so $6 \sqrt{6 y - 5}$ is on the left side of the inequality.

$4 + 6 \sqrt{6 y - 5} + 6 y < 9 y + 19$

Step 4.2

Move all terms not containing $6 \sqrt{6 y - 5}$ to the right side of the inequality.

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Step 4.2.1

Subtract $4$ from both sides of the inequality.

$6 \sqrt{6 y - 5} + 6 y < 9 y + 19 - 4$

Step 4.2.2

Subtract $6 y$ from both sides of the inequality.

$6 \sqrt{6 y - 5} < 9 y + 19 - 4 - 6 y$

Step 4.2.3

Subtract $6 y$ from $9 y$ .

$6 \sqrt{6 y - 5} < 3 y + 19 - 4$

Step 4.2.4

Subtract $4$ from $19$ .

$6 \sqrt{6 y - 5} < 3 y + 15$

Step 5

To remove the radical on the left side of the inequality, square both sides of the inequality.

${(6 \sqrt{6 y - 5})}^{2} < {(3 y + 15)}^{2}$

Step 6

Simplify each side of the inequality.

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 y - 5}$ as ${(6 y - 5)}^{\frac{1}{2}}$ .

${(6 {(6 y - 5)}^{\frac{1}{2}})}^{2} < {(3 y + 15)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(6 {(6 y - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $6 {(6 y - 5)}^{\frac{1}{2}}$ .

$6^{2} {({(6 y - 5)}^{\frac{1}{2}})}^{2} < {(3 y + 15)}^{2}$

Step 6.2.1.2

Raise $6$ to the power of $2$ .

$36 {({(6 y - 5)}^{\frac{1}{2}})}^{2} < {(3 y + 15)}^{2}$

Step 6.2.1.3

Multiply the exponents in ${({(6 y - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$36 {(6 y - 5)}^{\frac{1}{2} \cdot 2} < {(3 y + 15)}^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

$36 {(6 y - 5)}^{\frac{1}{2} \cdot 2} < {(3 y + 15)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$36 {(6 y - 5)}^{1} < {(3 y + 15)}^{2}$

Step 6.2.1.4

Simplify.

$36 (6 y - 5) < {(3 y + 15)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$36 (6 y) + 36 \cdot - 5 < {(3 y + 15)}^{2}$

Step 6.2.1.6

Multiply.

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Step 6.2.1.6.1

Multiply $6$ by $36$ .

$216 y + 36 \cdot - 5 < {(3 y + 15)}^{2}$

Step 6.2.1.6.2

Multiply $36$ by $- 5$ .

$216 y - 180 < {(3 y + 15)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(3 y + 15)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(3 y + 15)}^{2}$ as $(3 y + 15) (3 y + 15)$ .

$216 y - 180 < (3 y + 15) (3 y + 15)$

Step 6.3.1.2

Expand $(3 y + 15) (3 y + 15)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$216 y - 180 < 3 y (3 y + 15) + 15 (3 y + 15)$

Step 6.3.1.2.2

Apply the distributive property.

$216 y - 180 < 3 y (3 y) + 3 y \cdot 15 + 15 (3 y + 15)$

Step 6.3.1.2.3

Apply the distributive property.

$216 y - 180 < 3 y (3 y) + 3 y \cdot 15 + 15 (3 y) + 15 \cdot 15$

Step 6.3.1.3

Simplify and combine like terms.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$216 y - 180 < 3 \cdot 3 y \cdot y + 3 y \cdot 15 + 15 (3 y) + 15 \cdot 15$

Step 6.3.1.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 6.3.1.3.1.2.1

Move $y$ .

$216 y - 180 < 3 \cdot 3 (y \cdot y) + 3 y \cdot 15 + 15 (3 y) + 15 \cdot 15$

Step 6.3.1.3.1.2.2

Multiply $y$ by $y$ .

$216 y - 180 < 3 \cdot 3 y^{2} + 3 y \cdot 15 + 15 (3 y) + 15 \cdot 15$

Step 6.3.1.3.1.3

Multiply $3$ by $3$ .

$216 y - 180 < 9 y^{2} + 3 y \cdot 15 + 15 (3 y) + 15 \cdot 15$

Step 6.3.1.3.1.4

Multiply $15$ by $3$ .

$216 y - 180 < 9 y^{2} + 45 y + 15 (3 y) + 15 \cdot 15$

Step 6.3.1.3.1.5

Multiply $3$ by $15$ .

$216 y - 180 < 9 y^{2} + 45 y + 45 y + 15 \cdot 15$

Step 6.3.1.3.1.6

Multiply $15$ by $15$ .

$216 y - 180 < 9 y^{2} + 45 y + 45 y + 225$

Step 6.3.1.3.2

Add $45 y$ and $45 y$ .

$216 y - 180 < 9 y^{2} + 90 y + 225$

Step 7

Solve for $y$ .

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Step 7.1

Rewrite so $y$ is on the left side of the inequality.

$9 y^{2} + 90 y + 225 > 216 y - 180$

Step 7.2

Move all terms containing $y$ to the left side of the inequality.

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Step 7.2.1

Subtract $216 y$ from both sides of the inequality.

$9 y^{2} + 90 y + 225 - 216 y > - 180$

Step 7.2.2

Subtract $216 y$ from $90 y$ .

$9 y^{2} - 126 y + 225 > - 180$

Step 7.3

Convert the inequality to an equation.

$9 y^{2} - 126 y + 225 = - 180$

Step 7.4

Add $180$ to both sides of the equation.

$9 y^{2} - 126 y + 225 + 180 = 0$

Step 7.5

Add $225$ and $180$ .

$9 y^{2} - 126 y + 405 = 0$

Step 7.6

Factor the left side of the equation.

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Step 7.6.1

Factor $9$ out of $9 y^{2} - 126 y + 405$ .

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Step 7.6.1.1

Factor $9$ out of $9 y^{2}$ .

$9 (y^{2}) - 126 y + 405 = 0$

Step 7.6.1.2

Factor $9$ out of $- 126 y$ .

$9 (y^{2}) + 9 (- 14 y) + 405 = 0$

Step 7.6.1.3

Factor $9$ out of $405$ .

$9 y^{2} + 9 (- 14 y) + 9 \cdot 45 = 0$

Step 7.6.1.4

Factor $9$ out of $9 y^{2} + 9 (- 14 y)$ .

$9 (y^{2} - 14 y) + 9 \cdot 45 = 0$

Step 7.6.1.5

Factor $9$ out of $9 (y^{2} - 14 y) + 9 \cdot 45$ .

$9 (y^{2} - 14 y + 45) = 0$

Step 7.6.2

Factor.

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Step 7.6.2.1

Factor $y^{2} - 14 y + 45$ using the AC method.

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Step 7.6.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $45$ and whose sum is $- 14$ .

$- 9, - 5$

Step 7.6.2.1.2

Write the factored form using these integers.

$9 ((y - 9) (y - 5)) = 0$

Step 7.6.2.2

Remove unnecessary parentheses.

$9 (y - 9) (y - 5) = 0$

Step 7.7

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 9 = 0$

$y - 5 = 0$

Step 7.8

Set $y - 9$ equal to $0$ and solve for $y$ .

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Step 7.8.1

Set $y - 9$ equal to $0$ .

$y - 9 = 0$

Step 7.8.2

Add $9$ to both sides of the equation.

$y = 9$

Step 7.9

Set $y - 5$ equal to $0$ and solve for $y$ .

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Step 7.9.1

Set $y - 5$ equal to $0$ .

$y - 5 = 0$

Step 7.9.2

Add $5$ to both sides of the equation.

$y = 5$

Step 7.10

The final solution is all the values that make $9 (y - 9) (y - 5) = 0$ true.

$y = 9, 5$

Step 8

Find the domain of $\sqrt{9 y + 19} - \sqrt{6 y - 5} - 3$ .

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Step 8.1

Set the radicand in $\sqrt{9 y + 19}$ greater than or equal to $0$ to find where the expression is defined.

$9 y + 19 \geq 0$

Step 8.2

Solve for $y$ .

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Step 8.2.1

Subtract $19$ from both sides of the inequality.

$9 y \geq - 19$

Step 8.2.2

Divide each term in $9 y \geq - 19$ by $9$ and simplify.

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Step 8.2.2.1

Divide each term in $9 y \geq - 19$ by $9$ .

$\frac{9 y}{9} \geq \frac{- 19}{9}$

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $9$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

$\frac{9 y}{9} \geq \frac{- 19}{9}$

Step 8.2.2.2.1.2

Divide $y$ by $1$ .

$y \geq \frac{- 19}{9}$

Step 8.2.2.3

Simplify the right side.

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Step 8.2.2.3.1

Move the negative in front of the fraction.

$y \geq - \frac{19}{9}$

Step 8.3

Set the radicand in $\sqrt{6 y - 5}$ greater than or equal to $0$ to find where the expression is defined.

$6 y - 5 \geq 0$

Step 8.4

Solve for $y$ .

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Step 8.4.1

Add $5$ to both sides of the inequality.

$6 y \geq 5$

Step 8.4.2

Divide each term in $6 y \geq 5$ by $6$ and simplify.

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Step 8.4.2.1

Divide each term in $6 y \geq 5$ by $6$ .

$\frac{6 y}{6} \geq \frac{5}{6}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $6$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{6 y}{6} \geq \frac{5}{6}$

Step 8.4.2.2.1.2

Divide $y$ by $1$ .

$y \geq \frac{5}{6}$

Step 8.5

The domain is all values of $y$ that make the expression defined.

$[\frac{5}{6}, \infty)$

Step 9

Use each root to create test intervals.

$y < \frac{5}{6}$

$\frac{5}{6} < y < 5$

$5 < y < 9$

$y > 9$

Step 10

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 10.1

Test a value on the interval $y < \frac{5}{6}$ to see if it makes the inequality true.

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Step 10.1.1

Choose a value on the interval $y < \frac{5}{6}$ and see if this value makes the original inequality true.

$y = 0$

Step 10.1.2

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

$\sqrt{9 (0) + 19} - \sqrt{6 (0) - 5} > 3$

Step 10.1.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 10.2

Test a value on the interval $\frac{5}{6} < y < 5$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $\frac{5}{6} < y < 5$ and see if this value makes the original inequality true.

$y = 3$

Step 10.2.2

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

$\sqrt{9 (3) + 19} - \sqrt{6 (3) - 5} > 3$

Step 10.2.3

The left side $3.1767787$ is greater than the right side $3$ , which means that the given statement is always true.

True

Step 10.3

Test a value on the interval $5 < y < 9$ to see if it makes the inequality true.

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Step 10.3.1

Choose a value on the interval $5 < y < 9$ and see if this value makes the original inequality true.

$y = 7$

Step 10.3.2

Replace $y$ with $7$ in the original inequality.

$\sqrt{9 (7) + 19} - \sqrt{6 (7) - 5} > 3$

Step 10.3.3

The left side $2.9726226$ is not greater than the right side $3$ , which means that the given statement is false.

False

Step 10.4

Test a value on the interval $y > 9$ to see if it makes the inequality true.

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Step 10.4.1

Choose a value on the interval $y > 9$ and see if this value makes the original inequality true.

$y = 12$

Step 10.4.2

Replace $y$ with $12$ in the original inequality.

$\sqrt{9 (12) + 19} - \sqrt{6 (12) - 5} > 3$

Step 10.4.3

The left side $3.08407489$ is greater than the right side $3$ , which means that the given statement is always true.

True

Step 10.5

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

$y < \frac{5}{6}$ False

$\frac{5}{6} < y < 5$ True

$5 < y < 9$ False

$y > 9$ True

$y < \frac{5}{6}$ False

$\frac{5}{6} < y < 5$ True

$5 < y < 9$ False

$y > 9$ True

Step 11

The solution consists of all of the true intervals.

$\frac{5}{6} < y < 5$ or $y > 9$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$\frac{5}{6} < y < 5 or y > 9$

Interval Notation:

$(\frac{5}{6}, 5) \cup (9, \infty)$

Step 13