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
≥ 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.
