Question 169654
There are two lines graphed, so we need to find the equations of those two lines first. From there, we can find the inequalities.



Line 1: Line going through points (0,4) and (4,0)



So let's find the equation of line 1



First let's find the slope of the line through the points *[Tex \LARGE \left(0,4\right)] and *[Tex \LARGE \left(4,0\right)]



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



{{{m=(0-4)/(4-0)}}} Plug in {{{y[2]=0}}}, {{{y[1]=4}}}, {{{x[2]=4}}}, and {{{x[1]=0}}}



{{{m=(-4)/(4-0)}}} Subtract {{{4}}} from {{{0}}} to get {{{-4}}}



{{{m=(-4)/(4)}}} Subtract {{{0}}} from {{{4}}} to get {{{4}}}



{{{m=-1}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(0,4\right)] and *[Tex \LARGE \left(4,0\right)] is {{{m=-1}}}



Now let's use the point slope formula:



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



{{{y-4=-1(x-0)}}} Plug in {{{m=-1}}}, {{{x[1]=0}}}, and {{{y[1]=4}}}



{{{y-4=-1x+-1(-0)}}} Distribute



{{{y-4=-1x+0}}} Multiply



{{{y=-1x+0+4}}} Add 4 to both sides. 



{{{y=-1x+4}}} Combine like terms. 



{{{y=-x+4}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(0,4\right)] and *[Tex \LARGE \left(4,0\right)] is {{{y=-x+4}}}




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Line 2: Line through the points (0,0) and (3,6)



So let's find the equation of line 2




First let's find the slope of the line through the points *[Tex \LARGE \left(0,0\right)] and *[Tex \LARGE \left(3,6\right)]



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



{{{m=(6-0)/(3-0)}}} Plug in {{{y[2]=6}}}, {{{y[1]=0}}}, {{{x[2]=3}}}, and {{{x[1]=0}}}



{{{m=(6)/(3-0)}}} Subtract {{{0}}} from {{{6}}} to get {{{6}}}



{{{m=(6)/(3)}}} Subtract {{{0}}} from {{{3}}} to get {{{3}}}



{{{m=2}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(0,0\right)] and *[Tex \LARGE \left(3,6\right)] is {{{m=2}}}



Now let's use the point slope formula:



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



{{{y-0=2(x-0)}}} Plug in {{{m=2}}}, {{{x[1]=0}}}, and {{{y[1]=0}}}



{{{y-0=2x+2(-0)}}} Distribute



{{{y-0=2x+0}}} Multiply



{{{y=2x+0+0}}} Add 0 to both sides. 



{{{y=2x+0}}} Combine like terms. 



{{{y=2x}}} Remove the trailing zero



{{{y=2x}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(0,0\right)] and *[Tex \LARGE \left(3,6\right)] is {{{y=2x}}}




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So the system of <b>equations</b>(we'll convert them into inequalities later) is:


{{{system(y=-x+4,y=2x)}}}




Now, notice how the shaded region is BELOW both of the lines. In other words, EVERY point in the shaded region has a y-coordinate that is less than a point on either line.



So this means that we can take {{{y=-x+4}}} and convert it to {{{y<-x+4}}} (since every point in this shaded region is below the line {{{y=-x+4}}}) and we can take {{{y=2x}}} and convert it to {{{y<2x}}} (since every point in this shaded region is below the line {{{y=2x}}})



So we then end up with the system of inequalities


{{{system(y<-x+4,y<2x)}}}



Note: If you are INCLUDING the boundaries, then you need to change the inequality signs from *[Tex \LARGE <]  to *[Tex \LARGE \le] like this:



{{{system(y<=-x+4,y<=2x)}}}