Question

determine whether the statement is true or false. if it is true, explain why it is true. if it is false, give an example to show why it is false. the solution set of the inequality ax + by + c ≤ 0 is either a left half - plane or a lower half - plane. true. because the right - hand side of the equation is less than or equal to zero, the graph will fall to the left and below 0. true. because the right - hand side of the equation is less than or equal to zero, the graph will fall to the left and below - c, the opposite of the constant. true. the graph of 2x + 3y + 6 ≤ 0 is the lower or left half - plane. false. the graph of - 2x - 3y + 6 ≤ 0 is the upper or right half - plane. false. the solution set of the inequality ax + by + c ≤ 0 is either a right half - plane or an upper half - plane.

Explanation:

Step1: Analyze general linear - inequality

The general form of a linear inequality is $ax + by + c\leq0$. The boundary line is $ax+by + c = 0$.
The solution set of the inequality $ax + by + c\leq0$ represents the region on one - side of the line $ax + by + c = 0$.
If $a>0$, solving $ax+by + c\leq0$ for $x$ gives $x\leq-\frac{b}{a}y-\frac{c}{a}$, which represents a left - half plane.
If $b>0$, solving $ax + by + c\leq0$ for $y$ gives $y\leq-\frac{a}{b}x-\frac{c}{b}$, which represents a lower - half plane.

Step2: Consider a counter - example

Let's consider the inequality $- 2x-3y + 6\leq0$. First, rewrite it in the slope - intercept form $y\geq-\frac{2}{3}x + 2$. This represents the upper - half plane. Also, if we rewrite it to solve for $x$, $x\geq-\frac{3}{2}y + 3$, which represents the right - half plane.

Answer:

False. The graph of $-2x - 3y+6\leq0$ is the upper or right half - plane.