Question

graph the inequality on the axes below.
$-5x + 3y \leq -12$

Explanation:

Step1: Rewrite the inequality in slope - intercept form ($y = mx + b$)

We start with the inequality $-5x + 3y\leq - 12$. First, we isolate $y$ on one side.
Add $5x$ to both sides of the inequality: $3y\leq5x - 12$.
Then divide each term by 3: $y\leq\frac{5}{3}x-4$.

Step2: Graph the boundary line

The boundary line is given by the equation $y=\frac{5}{3}x - 4$. Since the inequality is $\leq$, the boundary line will be a solid line (because the points on the line are included in the solution set).

• Find the y - intercept: When $x = 0$, $y=\frac{5}{3}(0)-4=-4$. So the line passes through the point $(0, - 4)$.
• Find another point using the slope: The slope $m=\frac{5}{3}$, which means for every 3 units we move to the right (increase in $x$ by 3), we move up 5 units (increase in $y$ by 5). Starting from $(0, - 4)$, if we move $x = 3$, then $y=-4 + 5=1$. So the line also passes through the point $(3,1)$.

Step3: Determine the region to shade

We test a point not on the line. A common test point is the origin $(0,0)$.
Substitute $x = 0$ and $y = 0$ into the original inequality $-5(0)+3(0)\leq - 12$?
Simplify the left - hand side: $0\leq - 12$? This is false.
Since the test point $(0,0)$ does not satisfy the inequality, we shade the region that does not contain the origin. In other words, we shade the region below the line $y=\frac{5}{3}x - 4$ (because the inequality is $y\leq\frac{5}{3}x - 4$ and the origin is above the line and does not satisfy the inequality).

(Note: Since this is a graphing problem, the final answer is the graph of the line $y = \frac{5}{3}x-4$ (solid line) with the region below the line shaded. If we were to describe the key features: the boundary line has a slope of $\frac{5}{3}$ and a y - intercept of - 4, and the solution region is below this solid line.)

Answer:

The graph consists of a solid line with slope $\frac{5}{3}$ and y - intercept - 4 (passing through points like $(0, - 4)$ and $(3,1)$) and the region below the line is shaded.