Answer:

4. −3<x<−1

5. y<2 or y>4

6. x≤−5 or x≥−1

7.  x<1/3 or x>1/2

8. −5≤r≤ -1/2

9. r<−4 or r> 5/2

Solution

4. x2+4x+3=0

(x+1)(x+3)=0(Factor left side of equation)

x+1=0 or x+3=0(Set factors equal to 0)

x=−1 or x=−3

Check intervals in between critical points. (Test values in the intervals to see if they work.)

x<−3(Doesn't work in original inequality)

−3<x<−1(Works in original inequality)

x>−1(Doesn't work in original inequality)

5.y2−6y+8>0

Let's find the critical points of the inequality.

y2−6y+8=0

(y−2)(y−4)=0(Factor left side of equation)

y−2=0 or y−4=0(Set factors equal to 0)

y=2 or y=4

Check intervals in between critical points. (Test values in the intervals to see if they work.)

y<2(Works in original inequality)

2<y<4(Doesn't work in original inequality)

y>4(Works in original inequality)

6, x2+6x≥−5

Let's find the critical points of the inequality.

x2+6x=−5

x2+6x−(−5)=−5−(−5)(Subtract -5 from both sides)

x2+6x+5=0

(x+1)(x+5)=0(Factor left side of equation)

x+1=0 or x+5=0(Set factors equal to 0)

x=−1 or x=−5

Check intervals in between critical points. (Test values in the intervals to see if they work.)

x≤−5(Works in original inequality)

−5≤x≤−1(Doesn't work in original inequality)

x≥−1(Works in original inequality)

7. 6x2−5x+1>0

Let's find the critical points of the inequality.

6x2−5x+1=0

(3x−1)(2x−1)=0(Factor left side of equation)

3x−1=0 or 2x−1=0(Set factors equal to 0)

x=

1

3

or x=

1

2

Check intervals in between critical points. (Test values in the intervals to see if they work.)

x<

1

3

(Works in original inequality)

1

3

<x<

1

2

(Doesn't work in original inequality)

x>

1

2

(Works in original inequality)

8. 2r2+11r≤−5

Let's find the critical points of the inequality.

2r2+11r=−5

2r2+11r−(−5)=−5−(−5)(Subtract -5 from both sides)

2r2+11r+5=0

(2r+1)(r+5)=0(Factor left side of equation)

2r+1=0 or r+5=0(Set factors equal to 0)

r=

−1

2

or r=−5