- #1

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## Homework Statement

X(2x+7)≤0

## Homework Equations

## The Attempt at a Solution

Is my answer correct?

X≤-7/2

X≤1

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- Thread starter banana_banana
- Start date

- #1

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X(2x+7)≤0

Is my answer correct?

X≤-7/2

X≤1

- #2

HallsofIvy

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Homework Helper

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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)

- #3

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- #4

dynamicsolo

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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

2x + 7 = 0 --> x = -7/2 . Since the inequality is "less than

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 [tex]-\frac{7}{2} \leq x \leq 0[/tex]

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

[tex]-\frac{7}{2} \leq x \leq 0[/tex]

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