Question 149449
{{{s/6-2/9=0}}} Start with the given equation.



{{{18(s/cross(6)-2/cross(9))=18(0)}}} Multiply both sides by the LCD 18 to clear out the fractions.



{{{3s-4=0}}} Distribute and multiply.



{{{3s=0+4}}} Add {{{4}}} to both sides.



{{{3s=4}}} Combine like terms on the right side.



{{{s=(4)/(3)}}} Divide both sides by {{{3}}} to isolate {{{s}}}.



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Answer:


So the answer is {{{s=4/3}}} 



Which approximates to {{{s=1.333}}} 



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2)


{{{-43<20-9v<=-7}}} Start with the given compound inequality.



{{{-43-20<-9v<=-7-20}}} Subtract {{{20}}} from all sides.



{{{-63<-9v<=-7-20}}} Combine like terms on the left side.



{{{-63<-9v<=-27}}} Combine like terms on the right side.



{{{(-63)/-9>v>=(-27)/-9}}} Divide all sides by -9. Note: dividing all sides of the inequality will flip the signs



{{{7>v>=3}}} Reduce.



{{{3<=v<7}}} Rearrange the inequality.

So the answer in interval notation is   <font size="8">[</font>*[Tex \LARGE \bf{3,7}]<font size="8">)</font>



Also, the answer in set-builder notation is  *[Tex \LARGE \left\{v\|3 \le v < 7\right\}]



Here's the graph of the solution set


{{{drawing(500,80,-2, 12,-10, 10,
number_line( 500, -2, 12 ,3),

blue(line(3,0,7,0)),
blue(line(3,0.30,7,0.30)),
blue(line(3,0.15,7,0.15)),
blue(line(3,-0.15,7,-0.15)),
blue(line(3,-0.30,7,-0.30)),
circle(7,0,0.25),
circle(7,0,0.20)

)}}} Graph of the solution set


Note:

There is a <b>closed</b> circle at {{{v=3}}} which means that we're including this value in the solution set

Also, there is an <b>open</b> circle at {{{v=7}}} which means that we're excluding this value from the solution set.



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3) 


{{{2x-5y=-12}}} Start with the given equation.



{{{2(-1)-5y=-12}}} Plug in {{{x=-1}}}.



{{{-2-5y=-12}}} Multiply.



{{{-5y=-12+2}}} Add {{{2}}} to both sides.



{{{-5y=-10}}} Combine like terms on the right side.



{{{y=(-10)/(-5)}}} Divide both sides by {{{-5}}} to isolate {{{y}}}.



{{{y=2}}} Reduce.


So the ordered pair is (-1,2)



Note: you can plug in the numbers x=-1 and y=2 into the equation to check your answer.




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4)



{{{7*x-2*y=-5}}} Start with the given equation


Let's find the x-intercept


To find the x-intercept, let y=0 and solve for x:

{{{7*x-2*(0)=-5}}} Plug in {{{y=0}}}


{{{7*x=-5}}} Simplify


{{{x=-5/7}}} Divide both sides by 7




So the x-intercept is *[Tex \Large \left(-\frac{5}{7},0\right)] (note: the x-intercept will always have a y-coordinate equal to zero)




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{{{7*x-2*y=-5}}} Start with the given equation


Now let's find the y-intercept


To find the y-intercept, let x=0 and solve for y:

{{{7*(0)-2*y=-5}}} Plug in {{{x=0}}}


{{{2*y=-5}}} Simplify


{{{x=-5/-2}}} Divide both sides by -2




{{{y=5/2}}} Reduce




So the y-intercept is *[Tex \Large \left(0,\frac{5}{2}\right)] (note: the y-intercept will always have a x-coordinate equal to zero)


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So we have these intercepts:

x-intercept: *[Tex \Large \left(-\frac{5}{7},0\right)]


y-intercept: *[Tex \Large \left(0,\frac{5}{2}\right)]





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5)




{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula.



{{{m=(1--2)/(1--4)}}} Plug in {{{y[2]=1}}}, {{{y[1]=-2}}}, {{{x[2]=1}}}, {{{x[1]=-4}}}, , 



{{{m=(3)/(1--4)}}} Subtract {{{-2}}} from {{{1}}} to get {{{3}}}



{{{m=(3)/(5)}}} Subtract {{{-4}}} from {{{1}}} to get {{{5}}}



So the slope of the line that goes through the points *[Tex \LARGE \left(-4,-2\right)] and *[Tex \LARGE \left(1,1\right)] is {{{m=3/5}}}