Since the product of the binomials obtained should be negative, as stated by the inequality symbol (x+6)(x-6) <0, the quantities represented by the binomials should have different signs. Then, we have two options.
Option 1
Option 2
Factors' Signs
x+6<0 and x-6>0
x+6>0 and x-6<0
Simplified Form
x< - 6 and x>6
x>- 6 and x<6
Solution Set
no solution
- 6
The solution set for the inequality is - 6points at -6 and 6, as the inequality symbols do not include the boundary values. Then, we shade the values between them.
b We can see that on the left-hand side of the inequality we have the difference of squares. We can start by factoring it to see what the inequality implies for these factors.
Since the product of the binomials obtained should be positive, as stated by the inequality symbol (x+3)(x-3) >0, the quantities represented by the binomials should have the same signs. We have the two options.
Option 1
Option 2
Factors' Signs
x+3>0 and x-3>0
x+3<0 and x-3<0
Simplified Form
x> - 3 and x>3
x< - 3 and x<3
Solution Set
x>3
x<- 3
The solution set for the inequality is then x< - 3 or x>3. To graph the solution set we need to use two open points at -3 and 3, as the inequality symbols do not include the boundary values. Then, we shade to the left of -3 and to the right of 3.
c Notice that the inequality is asking for numbers whose square is less than - 4. x^2 < - 4
As any negative number becomes positive when squared and positive number remain positive, there are no numbers satisfying this inequality.
d We can start by factoring the expression on the left-hand side to see the implications of the inequality for the factors.
x^2 - 3x - 18 >0
To factor the expression on the left-hand side, we need to find two numbers such that their product is- 18 and their sum is- 3. Since - 18<0, both factors should have different sign. Also, since - 3 < 0, the greater factor should be negative. Let's check for these requirements for different factors of 8.
Factors of - 18
Sum
1,- 18
1-8 = - 7 *
2,- 9
2 - 9 = -7 *
3,- 6
3-6 = - 3 ✓
With this in mind, we can factor our expression.
x^2-3x-18 >0
⇕
(x + 3)(x - 6) >0
Since the product of the binomials obtained should be positive, as stated by the inequality symbol (x+3)(x-6) >0, the quantities represented by the binomials should have the same signs. Then, we have two two options.
Option 1
Option 2
Factors' Signs
x+3>0 and x-6>0
x+3<0 and x-6<0
Simplified Form
x> - 3 and x>6
x< - 3 and x<6
Solution Set
x>6
x<- 3
The solution set for the inequality is then x< - 3 or x>6. To graph the solution set we need to use two open points at -3 and 6, as the inequality symbols do not include the boundary values. Then, we shade to the left of -3 and to the right of 6.
