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By this equation, first the both negative signs cancel with each-other then 39/13, after that we'll get the 3 answer.

Betty goes to the store to get flour and sugar.
The amount of flour she buys, in pounds, is at least 6 pounds more than half the amount of sugar,
and is no more than twice the amount of sugar.
Find the least number of pounds of sugar that Betty could buy.

Let \(y\) = flour
Let \(x\) = sugar

\(\begin{array}{|rcll|} \hline y &\ge& 6+\dfrac x2 \\ y &\le& 2x \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline && & y \ge 6+\dfrac x2 \\ && & y \le 2x \\\\ \underline{x_{\text{min}}:} \\ 2x &=& 6+\dfrac x2 \\\\ 2x-\dfrac x2 &=& 6 \\\\ \dfrac32\cdot x &=& 6 \\\\ x &=& \dfrac23 \cdot 6 \\\\ \mathbf{ x} & \mathbf{=} & \mathbf{4\ \text{pounds}} \\ \hline \end{array}\)

The least number of pounds of sugar that Betty could buy are \(\mathbf{4\ \text{pounds}}\).

A single penalty kick is just one player taking a penalty against one goalkeeper.

There are 19 players who take a penalty kick against goalkeeper number one (that's 19 penalty kicks).

There are 19 players who take a penalty kick against goalkeeper number two (that's another 19 penalty kicks).

There are 19 players who take a penalty kick against goalkeeper number three (that's yet another 19 penalty kicks).

So how many in total?

.

Find all values of x such that 

\(\dfrac{x}{x+4} = -\dfrac{9}{x+3}\)

\dfrac{x}{x+4} = -\dfrac{9}{x+3}.
If you find more than one value, then list your solutions in increasing order, separated by commas.

\(\text{ $x\ne -4$ and $x\ne -3$ }\)

\(\begin{array}{|rcll|} \hline \dfrac{x}{x+4} &=& -\dfrac{9}{x+3} \quad & | \quad \cdot (x+3) \\\\ \dfrac{x(x+3)}{x+4} &=& -9 \quad & | \quad \cdot (x+4) \\\\ x(x+3) &=& -9(x+4) \\\\ x^2+3x &=& -9x -36 \quad & | \quad +9x \\\\ x^2+12x &=& -36 \quad & | \quad +36 \\\\ x^2+12x+36 &=&0 \\\\ x &=& \dfrac{-12\pm \sqrt{12^2 -4\cdot 36 } }{2} \\\\ x &=& \dfrac{-12\pm \sqrt{144 -144 } }{2} \\\\ x &=& \dfrac{-12\pm \sqrt{0} }{2} \\\\ x &=& \dfrac{-12\pm 0 }{2} \\\\ x &=& \dfrac{-12}{2} \\\\ \mathbf{x} &\mathbf{=}& \mathbf{-6} \\ \hline \end{array}\)

What values of x satisfy |x - 4| + |x + 4| <= 10 ?

Express your answer in interval notation.

\( \huge |x + 4| + |x - 4| \le 10\)

Here we have two different amounts.

To dissolve them, we must make a double case distinction.

Usually we did this one after the other.
For reasons of space, we start with the first case in which the content of the left amount
is greater than or equal to zero.

\(\begin{array}{|rcll|} \hline & \underline{x\ge -4:}& \\\\ & x+4 + |x-4 | \le 10 \\ \\ \underline{\text{for } x\ge 4:} && \underline{\text{for } x \lt 4: } \\\\ x+4+x-4 \le 10 && x+4-(x-4) \le 10 \\ 2x \le 10 && 8 \le 10 \\ \boxed{ x \le 5 } && \\ \hline \end{array} \)

The other case was \(x \lt -4\).
In this area the content of the right amount is
always negative. A further case distinction is therefore not necessary here.

\(\begin{array}{|rcll|} \hline & \underline{x \lt -4:} \\\\ & -(x+4) - (x-4) \le 10 \\ \\ & -x-4-x+4 \le 10 \\ & -2x \le 10 \quad & \quad :(-2)\\ & x \ge \dfrac{10}{-2} \\ & \boxed{x \ge -5} \\ \hline \end{array} \)

Values of x in interval notation: [-5 , 5  ]

I think 36,000 ways, hint: what are the total arrangements?

Both the negative signs cancel out, so we are left with 39/13=3.

-4 is the answer

-9/8

-3/2, or -1.5

Here is another way:

1304/24 =54 1/3 miles - average speed of both men
[54 1/3 x 2] / 1.75 =~62 miles - average speed of Vlado
62.1 x 0.75 =~46.7 miles - average speed of Santi
12 hours x 62 =~744 - miles driven by Vlado.
12 hours x 46.7 =~560 - miles driven by Santi.

What didn't work  ??

It didn't work?

(-4, 9)...   10  left  7 down  puts her at  (  -4 - 10 , 9  - 7)  = ( -14, 2)

So....  the points  ( -4, 9)  and (-14,2)  are on the  line

Slope of this line  =   [ 2 - 9] / [ -14 - -4 ]  =  [ -7] / [ -10]  =  7/10

Equation of the line

y  = (7/10)(x - - 4)  + 9

y  = (7/10) ( x + 4)  + 9     multiply through by 10

10y = 7 (x + 4)  + 90

10y = 7x + 28  + 90

-7x + 10y  = 118

Here's a graph : https://www.desmos.com/calculator/jlpnh4ai3p

2x + 7  = -5      

2x  = -12

x = -6

-x^2 - x + 1    = -5     multiply through by -1

x^2 + x - 1  = 5

x^2 + x  - 6  = 0   factor

(x + 3) ( x - 2)  =  0

Set each factor to  0  and solve for  x and we get   x  = -3   (out of the domain for the function)  or  x = 2

So...the sum of the  x values such that f(x)   = -5  ⇒   -6  + 2   =  -4 

2, 4 , 6 , 8, 10

Yep...taking turns driving is the way I interpret  it...each drives 12 hours

Thaaank you, u helped big time, BIG TIME!

So each drove  12 hrs

And Rate * Time = Distance

So...call Vlado's rate R  and Santi's rate (.75R)    and we have

Santi's Rate * Time Driving  + Vlado's Rate  * Time Driving  = Total Distance

.75R (12)   + R(12)  = 1304  simplify

R(9) + R(12)  =  1304

R (21)  = 1304   divide both sids by 21

R ≈ 62.1  mph

So Santi  drove  ≈   .75 (62.1) (12)  ≈ 559 miles ⇒  round to   560 miles

See  under my original answer ...it won't change anything, much...

Here's a corrected graph : https://www.desmos.com/calculator/rqfmbx0gfb

oh ok thank you!

It wouldn't change the answer, much....the graph of  x + y ≤ 4  would be a solid line instead of the dashed line of  x + y < 4

The answer for the chosen points would be the same...

wait actually the second equation was suppose to have a ≤

Thank you!!!

OK...easy enough....

Here's the desmos graph : https://www.desmos.com/calculator/h3ywx9slgp

Note  that the point  ( 0,0)  is a solution because   0 > -2   and  0 + 0  < 4

Note that the point  (0,5)  is not a solution  because  5 > -2   but   0 + 5  is  not less than 4