How do you graph y <2x+4 and -3x-2y>=6?

Jun 1, 2018

See explanation

Explanation:

Given:

$y < 2 x + 4 \text{ } \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . E q n \left(1\right)$

$- 3 x - 2 y \ge 6 \text{ } \ldots \ldots \ldots \ldots \ldots \ldots \ldots E q n \left(2\right)$

$\textcolor{b l u e}{\text{Converting "Eqn(2)" into standardised form }}$

Consider $E q n \left(2\right)$

Add $2 y$ to both sides and subtract $6$ from both sides giving:

$- 3 x - 6 \ge 2 y$

Divide both sides by 2

$- \frac{3}{2} x - \frac{6}{2} \ge y$

Write the order as per convention

$y \le - \frac{3}{2} x - 3 \text{ } \ldots \ldots \ldots . E q n \left({2}_{a}\right)$
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Plot them $\textcolor{m a \ge n t a}{\underline{\text{as if they were}}}$
y=-3/2 x-3 color(white)("dddd") ->color(white)("dddd")"really is: " y<2x+4 " "Eqn(2_a)

Feasible solution area is below color(white)()"and "ul(color(red)("including the solid line"))color(white)("d") Eqn(2_a)

and:

y=2x+4 color(white)("dddd") ->color(white)("dddd")"really is: " y<2x+4" "Eqn(1)

Feasible solution area is below and
$\textcolor{red}{\underline{\text{excluding the dotted line }}} E q n \left(1\right)$

The solution area is where these two are coincidental (coincide).
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine key point - Intersection of the two lines}}$

$y = 2 x + 4 \text{ } \ldots \ldots \ldots \ldots E q n \left(1\right)$
$y = - \frac{3}{2} x - 3 \text{ } \ldots \ldots E q n \left({2}_{a}\right)$

$E q n \left(1\right) - E q n \left(2\right)$ to 'get rid' of the $y ' s$

$0 = \frac{7}{2} x + 7$

$\textcolor{g r e e n}{x = - 7 \times \frac{2}{7} = - 2}$

Substitute $x = - 2$ in $E q n \left({2}_{a}\right)$ giving:

$y = \left(- \frac{3}{2}\right) \left(- 2\right) - 3 = + 3 - 3 = 0$

color(blue)(ul(bar(| color(white)(2/2) "Intersection"->(x,y)=(-2,0)color(white)(2/2) |))
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine key point - axis intercepts for } y < 2 x + 4}$

Set $y = 2 x + 4$

Set x=0 -> color(blue)(bar(ul(|color(white)(2/2)y_("intercept")=4color(white)(2/2)|))

Set y=0 -> color(blue)(ul(bar(|color(white)(2/2) x_("intercept")=-2color(white)(2/2)|))
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine key point - axis intercepts for } - 3 x - 2 y \ge 6}$

$E q n \left({2}_{a}\right) \to y \le - \frac{3}{2} x - 3$

Set $y = - \frac{3}{2} x - 3$

Set $x = 0 \to \textcolor{b l u e}{\underline{\overline{| \textcolor{w h i t e}{\frac{2}{2}} {y}_{\text{intercept}} = - 3 \textcolor{w h i t e}{\frac{2}{2}} |}}}$

Set $y = 0 \to \textcolor{b l u e}{\underline{\overline{| \textcolor{w h i t e}{\frac{2}{2}} {x}_{\text{intercept}} = - 2 \textcolor{w h i t e}{\frac{2}{2}} |}}}$