What is the smallest possible integer value of x for which (x-8)(x+5)<0?
a) -8
b)-7
c)-6
d)-5
e)-4
If x + 5 < 0, then -5 would be the integer.
If x -8 < 0, then 8 would be the integer.
Which is smaller?
a
To find the smallest possible integer value of x for which (x-8)(x+5)<0, we need to determine when the expression changes sign.
The expression (x-8)(x+5) changes sign when one of the factors changes sign. In other words, either (x-8)<0 and (x+5)>0, or (x-8)>0 and (x+5)<0.
To solve the first case, we set (x-8) < 0 and solve for x:
x - 8 < 0
x < 8
Next, we solve the second case:
(x+5) > 0
x > -5
Now, we are looking for the smallest possible integer value of x that satisfies both conditions: x < 8 and x > -5.
The smallest integer value that satisfies both conditions is -4.
Therefore, the answer is e) -4.
To find the smallest possible integer value of x for which (x-8)(x+5)<0, we need to determine the range of values for which the expression (x-8)(x+5) is negative.
First, let's analyze the expression (x-8)(x+5). This expression is negative when either one of the factors is negative and the other is positive.
So we have two cases to consider:
Case 1: (x-8) < 0 and (x+5) > 0
To solve the inequality (x-8) < 0, we add 8 to both sides: x < 8.
To solve the inequality (x+5) > 0, we subtract 5 from both sides: x > -5.
Therefore, in this case, x lies in the range -5 < x < 8.
Case 2: (x-8) > 0 and (x+5) < 0
To solve the inequality (x-8) > 0, we add 8 to both sides: x > 8.
To solve the inequality (x+5) < 0, we subtract 5 from both sides: x < -5.
Therefore, in this case, there are no solutions since the two inequalities cannot be simultaneously true.
Now we need to find the smallest integer value of x that falls in the range -5 < x < 8. The smallest integer within this range is -4.
Therefore, the answer is e) -4.