Simplify 3<=3-9y+24-4+y^2

$3 \leq 3 - 9 y + 24 - 4 + y^{2}$

Step 1

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

$3 - 9 y + 24 - 4 + y^{2} \geq 3$

Step 2

Simplify $3 - 9 y + 24 - 4 + y^{2}$ .

Add $3$ and $24$ .

$- 9 y + 27 - 4 + y^{2} \geq 3$

Subtract $4$ from $27$ .

$- 9 y + 23 + y^{2} \geq 3$

Step 3

Convert the inequality to an equation.

$- 9 y + 23 + y^{2} = 3$

Step 4

Subtract $3$ from both sides of the equation.

$- 9 y + 23 + y^{2} - 3 = 0$

Step 5

Subtract $3$ from $23$ .

$- 9 y + y^{2} + 20 = 0$

Step 6

Factor the left side of the equation.

Let $u = y$ . Substitute $u$ for all occurrences of $y$ .

$- 9 u + u^{2} + 20 = 0$

Factor $- 9 u + u^{2} + 20$ using the AC method.

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 $20$ and whose sum is $- 9$ .

$- 5, - 4$

Write the factored form using these integers.

$(u - 5) (u - 4) = 0$

Replace all occurrences of $u$ with $y$ .

$(y - 5) (y - 4) = 0$

Step 7

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

$y - 5 = 0$

$y - 4 = 0$

Step 8

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

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

$y - 5 = 0$

Add $5$ to both sides of the equation.

$y = 5$

Step 9

Set $y - 4$ equal to $0$ and solve for $y$ .

Set $y - 4$ equal to $0$ .

$y - 4 = 0$

Add $4$ to both sides of the equation.

$y = 4$

Step 10

The final solution is all the values that make $(y - 5) (y - 4) = 0$ true.

$y = 5, 4$

Step 11

Use each root to create test intervals.

$y < 4$

$4 < y < 5$

$y > 5$

Step 12

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

Test a value on the interval $y < 4$ to see if it makes the inequality true.

Choose a value on the interval $y < 4$ and see if this value makes the original inequality true.

$y = 0$

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

$3 \leq 3 - 9 \cdot 0 + 24 - 4 + {(0)}^{2}$

The left side $3$ is less than the right side $23$ , which means that the given statement is always true.

True

Test a value on the interval $4 < y < 5$ to see if it makes the inequality true.

Choose a value on the interval $4 < y < 5$ and see if this value makes the original inequality true.

$y = 4.5$

Replace $y$ with $4.5$ in the original inequality.

$3 \leq 3 - 9 \cdot 4.5 + 24 - 4 + {(4.5)}^{2}$

The left side $3$ is greater than the right side $2.75$ , which means that the given statement is false.

False

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

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

$y = 8$

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

$3 \leq 3 - 9 \cdot 8 + 24 - 4 + {(8)}^{2}$

The left side $3$ is less than the right side $15$ , which means that the given statement is always true.

True

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

$y < 4$ True

$4 < y < 5$ False

$y > 5$ True

$y < 4$ True

$4 < y < 5$ False

$y > 5$ True

Step 13

The solution consists of all of the true intervals.

$y \leq 4$ or $y \geq 5$

Step 14

The result can be shown in multiple forms.

Inequality Form:

$y \leq 4 or y \geq 5$

Interval Notation:

$(- \infty, 4] \cup [5, \infty)$

Step 15

Simplify 3<=3-9y+24-4+y^2

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