x2+x≥6

Convert the inequality to an equation.

x2+x=6

Move 6 to the left side of the equation by subtracting it from both sides.

x2+x-6=0

Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is -6 and whose sum is 1.

-2,3

Write the factored form using these integers.

(x-2)(x+3)=0

(x-2)(x+3)=0

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

x-2=0

x+3=0

Set the first factor equal to 0.

x-2=0

Add 2 to both sides of the equation.

x=2

x=2

Set the next factor equal to 0.

x+3=0

Subtract 3 from both sides of the equation.

x=-3

x=-3

The final solution is all the values that make (x-2)(x+3)=0 true.

x=2,-3

Use each root to create test intervals.

x<-3

-3<x<2

x>2

Test a value on the interval x<-3 to see if it makes the inequality true.

Choose a value on the interval x<-3 and see if this value makes the original inequality true.

x=-6

Replace x with -6 in the original inequality.

(-6)2-6≥6

The left side 30 is greater than the right side 6, which means that the given statement is always true.

True

True

Test a value on the interval -3<x<2 to see if it makes the inequality true.

Choose a value on the interval -3<x<2 and see if this value makes the original inequality true.

x=0

Replace x with 0 in the original inequality.

(0)2+0≥6

The left side 0 is less than the right side 6, which means that the given statement is false.

False

False

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

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

x=4

Replace x with 4 in the original inequality.

(4)2+4≥6

The left side 20 is greater than the right side 6, which means that the given statement is always true.

True

True

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

x<-3 True

-3<x<2 False

x>2 True

x<-3 True

-3<x<2 False

x>2 True

The solution consists of all of the true intervals.

x≤-3 or x≥2

The result can be shown in multiple forms.

Inequality Form:

x≤-3 or x≥2

Interval Notation:

(-∞,-3]∪[2,∞)

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