Homework help

[banana_banana]

1. The problem statement, all variables and given/known data


X(2x+7)≤0

2. Relevant equations



3. The attempt at a solution
Is my answer correct?

X≤-7/2
X≤1

[HallsofIvy]

Can't you check it yourself? x= -4< -7/2. if x= -4, then x(2x+7)= (-4)(-8+ 7)= -4(-1)= 4 which is NOT less than 0. I can see that x= -7/2 makes x(2x+7)= 0. Where did you get x= 1?

The graph of y= x(2x+ 7) is a parabola. It crosses the x-axis where x(2x+7)= 0. What are the roots of that? Which part of the parabola is below the x-axis? If it is not obvious just check that value at some points: say x= -4, x= -1, and x= 1 (-4< -7/2< -1< 0< 1)

[banana_banana]

but if X≤-7/2 then it is like this ((-)infinite, -7/20] what about the other interval notation to be able to have union if x will be transposed to the other side of zero.

[dynamicsolo]

The set x < -7/2 couldn't be part of the solution in any case because that would make both of the terms in the product x·(2x+7) negative, and thus the product would be positive (as Halls has discussed).

When solving a product inequality, you need to examine cases. There are two approaches that are customarily taught (without resorting to graphing). You would solve the equation , x·(2x+7) = 0 , first in order to find the boundaries of the possible intervals which solve the inequality. Here, we would have either x = 0 or
2x + 7 = 0 --> x = -7/2 . Since the inequality is "less than or equal to", these values will be included in the solution; if the inequality were, say, just "less than", we would omit them, but keep them in mind.

One approach then uses these points to set the boundaries of the intervals:

x < -7/2 , -7/2 < x < 0 , x > 0

You would then choose a number in each interval and see what the sign of the result is (pick something easy to calculate). We find:

x = -4: (-4)·[2(-4) + 7] = (-4)(-1) = 4 > 0

x = -1: (-1)·[2(-1) + 7] = (-1)(5) = -5 < 0

x = +1: (1)·[2(1) + 7] = (1)(9) = 9 > 0

Only the number between -7/2 and 0 works in the inequality, so the one solution interval for this inequality is -\frac{7}{2} \leq x \leq 0

The other approach is more analytical and involves looking at the signs of the factors. We start from the inequality x·(2x+7) < 0 and consider that there are two ways for the product to be negative. The two factors would have to have opposite signs, so either

x > 0 AND 2x + 7 < 0 ,
which means x > 0 AND x < -7/2 , which is impossible

OR

x < 0 AND 2x + 7 > 0 ,
which means x < 0 AND x > -7/2 , which can happen and requires the interval
-7/2 < x < 0

We must also include the points x = -7/2 and x = 0 themselves, since they satisfy the equation portion of our inequality, giving us the solution interval

-\frac{7}{2} \leq x \leq 0