Algebra Examples

$2 y + 2 y \geq 4 y + 7 y + 2 + 8 + y^{2}$

Step 1

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

$4 y + 7 y + 2 + 8 + y^{2} \leq 2 y + 2 y$

Step 2

Simplify $4 y + 7 y + 2 + 8 + y^{2}$ .

Tap for more steps...

Step 2.1

Add $4 y$ and $7 y$ .

$11 y + 2 + 8 + y^{2} \leq 2 y + 2 y$

Step 2.2

Add $2$ and $8$ .

$11 y + 10 + y^{2} \leq 2 y + 2 y$

Step 3

Add $2 y$ and $2 y$ .

$11 y + 10 + y^{2} \leq 4 y$

Step 4

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

Tap for more steps...

Step 4.1

Subtract $4 y$ from both sides of the inequality.

$11 y + 10 + y^{2} - 4 y \leq 0$

Step 4.2

Subtract $4 y$ from $11 y$ .

$7 y + 10 + y^{2} \leq 0$

Step 5

Convert the inequality to an equation.

$7 y + 10 + y^{2} = 0$

Step 6

Factor $7 y + 10 + y^{2}$ using the AC method.

Tap for more steps...

Step 6.1

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 $10$ and whose sum is $7$ .

$2, 5$

Step 6.2

Write the factored form using these integers.

$(y + 2) (y + 5) = 0$

Step 7

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

$y + 2 = 0$

$y + 5 = 0$

Step 8

Set $y + 2$ equal to $0$ and solve for $y$ .

Tap for more steps...

Step 8.1

Set $y + 2$ equal to $0$ .

$y + 2 = 0$

Step 8.2

Subtract $2$ from both sides of the equation.

$y = - 2$

Step 9

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

Tap for more steps...

Step 9.1

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

$y + 5 = 0$

Step 9.2

Subtract $5$ from both sides of the equation.

$y = - 5$

Step 10

The final solution is all the values that make $(y + 2) (y + 5) = 0$ true.

$y = - 2, - 5$

Step 11

Use each root to create test intervals.

$y < - 5$

$- 5 < y < - 2$

$y > - 2$

Step 12

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

Tap for more steps...

Step 12.1

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

Tap for more steps...

Step 12.1.1

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

$y = - 8$

Step 12.1.2

Replace $y$ with $- 8$ in the original inequality.

$2 (- 8) + 2 (- 8) \geq 4 (- 8) + 7 (- 8) + 2 + 8 + {(- 8)}^{2}$

Step 12.1.3

The left side $- 32$ is less than the right side $- 14$ , which means that the given statement is false.

False

Step 12.2

Test a value on the interval $- 5 < y < - 2$ to see if it makes the inequality true.

Tap for more steps...

Step 12.2.1

Choose a value on the interval $- 5 < y < - 2$ and see if this value makes the original inequality true.

$y = - 4$

Step 12.2.2

Replace $y$ with $- 4$ in the original inequality.

$2 (- 4) + 2 (- 4) \geq 4 (- 4) + 7 (- 4) + 2 + 8 + {(- 4)}^{2}$

Step 12.2.3

The left side $- 16$ is greater than the right side $- 18$ , which means that the given statement is always true.

True

Step 12.3

Test a value on the interval $y > - 2$ to see if it makes the inequality true.

Tap for more steps...

Step 12.3.1

Choose a value on the interval $y > - 2$ and see if this value makes the original inequality true.

$y = 0$

Step 12.3.2

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

$2 (0) + 2 (0) \geq 4 (0) + 7 (0) + 2 + 8 + {(0)}^{2}$

Step 12.3.3

The left side $0$ is less than the right side $10$ , which means that the given statement is false.

False

Step 12.4

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

$y < - 5$ False

$- 5 < y < - 2$ True

$y > - 2$ False

$y < - 5$ False

$- 5 < y < - 2$ True

$y > - 2$ False

Step 13

The solution consists of all of the true intervals.

$- 5 \leq y \leq - 2$

Step 14

The result can be shown in multiple forms.

Inequality Form:

$- 5 \leq y \leq - 2$

Interval Notation:

$[- 5, - 2]$

Step 15