Question

Write the inequality for the following graph.

• Do not include spaces in your answer.
• Write fractions inside parentheses using the "/" symbol.
1
o For example, write y < x-7 as y<(1/2)x-7.

Answer 1

so from the picture below, let's before we get the INEQUALITY, let's first get its EQUALITY, or namely the equation of the line, so hmm for that we'd need to points from the line itself, let's see hmm the points (-1 , 8) and (4 , -7) are on it, and that's all we need to get the equation

`(\stackrel{x_1}{-1}~,~\stackrel{y_1}{8})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{-7}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-7}-\stackrel{y1}{8}}}{\underset{\textit{\large run}} {\underset{x_2}{4}-\underset{x_1}{(-1)}}} \implies \cfrac{ -15 }{4 +1} \implies \cfrac{ -15 }{ 5 } \implies -3`

`\begin{array}ll \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{8}=\stackrel{m}{-3}(x-\stackrel{x_1}{(-1)}) \implies y -8 = -3 ( x +1) \\\\\\ y -8 = -3 x -3 \implies y = -3 x +5`

so that's its equation, now let's work on the dashing and shading.

so is dashed, meaning the borderline points are not included, that will mean either > or <, let me give you the cheap answer for the shading, hmm the shading is up, so its really using ">", so y > -3x + 5.

Now, let me give you the not-so-cheap answer for the shading.

we know the top is shaded, so that is the TRUE REGION, and the non-shaded part is the FALSE REGION. On the True region the inequality is true and on False is false, no surprises there, so if we want to know which one to use either > or <, we can say hmm we have y = -3x + 5, ok, let's pick a point on the True region, say hmmm (4 , 10), that's on the shaded part so

`y = -3x+5\implies \begin{cases} x=4\\ y=10 \end{cases}\implies 10=-3(4)+5 \\\\\\ 10=-12+5\implies 10\\e -7\implies 10 \stackrel{ \stackrel{\textit{\small notice}}{\downarrow } }{ > } -7`

since we know 10 > 7, then ">" makes that statement TRUE, then we know what we need is really ">".

y > -3x + 5