what is the graph of the linear inequality 2x-3y<12

Accepted Solution

Answer:We understand a linear inequality as an inequality involving a linear function. It's important to know that a linear inequality contains one of the symbols of inequality:< less than > greater than ≀ less than or equal to β‰₯ greater than or equal to In this problem, we have:[tex]2x-3y<12[/tex]In this case, we use the symbol <, so this indicates that [tex]2x-3y[/tex] is less than 12. To solve this, let's write the linear equation in slope-intercept form:Step 1: Write the expression as an equation:[tex]2x-3y=12[/tex]Step 2: Subtract -2x from both sides:[tex]2x-3y-2x=12-2x \\ \\ -3y=12-2x[/tex]Step 3: Multiply the entire equation by -1/3[tex]-\frac{1}{3}(-3y)=-\frac{1}{3}(12-2x) \\ \\ y=\frac{2}{3}x-4[/tex]The graph of this equation is shown in the firs figure below. To know what's the shaded region let's take point (0, 0) and test it in the inequality:[tex]2(0)-3(0)<12 \\ \\ 0<12 \ TRUE![/tex]Since this is true, the shaded region includes point (0, 0) and this is above the graph. We have to draw a dotted line since equality is not included in the solution and this is shown in the second figure below.