Brainliest if correct

For what values of $x$ is $$\frac{x^2 + x + 3}{2x^2 + x - 6} \ge 0?$$ note: be thorough and explain why all points in your answe

Rus_ich [418]

To solve the inequality $\frac{x^2+x+3}{2x^2+x-6}\ge \:0$

$\mathrm{Factor\:the\:left\:hand\:side\:}\frac{x^2+x+3}{2x^2+x-6}:$

The numerator $x^2+x+3$ is not factorizable.

so factor the denominator $2x^2+x-6$ :

$2x^2+x-6=\left(2x^2-3x\right)+\left(4x-6\right)=x\left(2x-3\right)+2\left(2x-3\right)\\ \mathrm{Factor\:out\:}x\mathrm{\:from\:}2x^2-3x\mathrm{:\quad }x\left(2x-3\right)\\ \mathrm{Factor\:out\:}2\mathrm{\:from\:}4x-6\mathrm{:\quad }2\left(2x-3\right)\\$

Now take $\mathrm{Factor\:out\:common\:term\:}\left(2x-3\right)$

Then we get factor of the denominator as $\left(2x-3\right)\left(x+2\right)$

Thus $\frac{x^2+x+3}{2x^2+x-6}=\frac{x^2+x+3}{\left(x+2\right)\left(2x-3\right)}$

Now $\mathrm{Compute\:the\:signs\:of\:the\:factors\:of\:}\frac{x^2+x+3}{\left(x+2\right)\left(2x-3\right)}\\$

Signs of $x^2+x+3>0$

$\mathrm{Choosing\:ranges\:that\:satisfy\:the\:required\:condition:}\:\ge \:\:0$

$x\frac{3}{2}$ is the required solution of the given inequality.

5 0

3 years ago

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Simplify the following expression.

Katyanochek1 [597]

Where is the expression?

3 0

3 years ago

Find the exact location of all the relative and absolute extrema of the function. HINT [See Examples 1 and 2.] (Order your answe

icang [17]

Answer:

(-1, -32) absolute minimum
(0, 0) relative maximum
(2, -32) absolute minimum
(+∞, +∞) absolute maximum (or "no absolute maximum")

Step-by-step explanation:

There will be extremes at the ends of the domain interval, and at turning points where the first derivative is zero.

The derivative is ...

h'(t) = 24t^2 -48t = 24t(t -2)

This has zeros at t=0 and t=2, so that is where extremes will be located.

We can determine relative and absolute extrema by evaluating the function at the interval ends and at the turning points.

h(-1) = 8(-1)²(-1-3) = -32

h(0) = 8(0)(0-3) = 0

h(2) = 8(2²)(2 -3) = -32

h(∞) = 8(∞)³ = ∞

The absolute minimum is -32, found at t=-1 and at t=2. The absolute maximum is ∞, found at t→∞. The relative maximum is 0, found at t=0.

The extrema are ...

(-1, -32) absolute minimum
(0, 0) relative maximum
(2, -32) absolute minimum
(+∞, +∞) absolute maximum

_____

Normally, we would not list (∞, ∞) as being an absolute maximum, because it is not a specific value at a specific point. Rather, we might say there is no absolute maximum.

5 0

2 years ago

What are the four kinds of parabolas

Leya [2.2K]

Answer: Focus, Vertex, Axis, and Directrix.

Step-by-step explanation:

6 0

2 years ago

the sum of the two trains is 723.5 mph if the speed of the first train is 12.5 mph faster than that of the second train find the

jonny [76]

First, let's see if we can rewrite this word problem a little bit more mathematically. We won't get to mathy or technical so no worries. We just want to look at it in a more straightforward way, if we can.

Train A's mph plus Train B's mph summed equal 723.5 mph. Train A's mph is greater than Train B's mph by 12.5 mph.
So what should we do to solve this problem? Since we are dealing with two of something and we know the value of the two combined, it might make sense to start by dividing that value by 2.
723.5 / 2 = <em /> 361.75. If the two trains were travelling at the same speed, we would be done. Unfortunately, they are not so we need to think about this some more.
Train A is going 12.5 mph faster than Train B. Let's rewrite.
Train A mph = 12.5 + 361.75 = 374.25 Okay, so Train A is travelling at a speed of 374.25 mph. So we're done right? Not exactly. We are asked to fing the speeds of BOTH trains. How do we find the speed of Train B? We have added a portion of the combined total to Train A. It seems to follow, then, we should probably subtract the same portion from Train A. What are we going to do? You guessed it! Rewrite.
Train B mph = 361.75 - 12.5 = 349.25 HA HA! We seem to have figured it out. Let's do one last thing to check our work. Let's add the two trains' speeds together. If we did this right, we should get our summed value of 723.5 mph
374.25 + 349.25 = 723.5
Pat yourself on the back! We did it!

374.25 + 349.25 =

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3 years ago

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