Inequalities In Slope-Intercept Form A Step-by-Step Guide

Hey guys! Today, we're diving into the world of inequalities and how to transform them into the oh-so-useful slope-intercept form. We'll break down a specific system of inequalities, step-by-step, to make sure you've got a solid grasp of this concept. So, let's get started!

The Given System of Inequalities

We're tackling the following system:

$$\begin{array}{l} 4x - 5y \leq 1 \\ \frac{1}{2}y - x \leq 3 \end{array}$$

Our mission? To rewrite these inequalities in slope-intercept form. Remember, slope-intercept form is your friend! It looks like this: y = mx + b, where m is the slope and b is the y-intercept. This form makes it super easy to visualize and graph linear inequalities. Now, let's get to work transforming our inequalities!

Transforming the First Inequality: 4x - 5y ≤ 1

First up, we've got 4x - 5y ≤ 1. Our goal is to isolate y on one side of the inequality. Here’s how we do it:

1. Subtract 4x from both sides: This gets us closer to isolating the y term. We now have -5y ≤ -4x + 1.
2. Divide both sides by -5: This is a crucial step! Remember, when you multiply or divide both sides of an inequality by a negative number, you need to flip the inequality sign. So, dividing by -5 gives us y ≥ (4/5)x - (1/5).

Why did we flip the sign? Imagine you have -1 < 2. If you multiply both sides by -1, you get 1 > -2. See how the inequality sign flipped to maintain the truth of the statement? It's the same principle here.

So, the first inequality in slope-intercept form is y ≥ (4/5)x - (1/5). Notice that the y is now isolated, and we can easily identify the slope (4/5) and the y-intercept (-1/5). This form tells us that y is greater than or equal to a line with a slope of 4/5 and a y-intercept of -1/5. This means we'll shade above this line when we graph it.

Transforming the Second Inequality: (1/2)y - x ≤ 3

Next, we'll tackle the second inequality: (1/2)y - x ≤ 3. Again, we want to isolate y. Let's break it down:

1. Add x to both sides: This moves the x term to the right side, giving us (1/2)y ≤ x + 3.
2. Multiply both sides by 2: To get rid of the fraction, we multiply both sides by 2. This gives us y ≤ 2x + 6.

That's it! The second inequality in slope-intercept form is y ≤ 2x + 6. Here, the slope is 2 and the y-intercept is 6. This inequality tells us that y is less than or equal to a line with a slope of 2 and a y-intercept of 6. When graphing, we'll shade below this line.

Putting It All Together: The Solution

Now we have both inequalities in slope-intercept form:

• y ≥ (4/5)x - (1/5)
• y ≤ 2x + 6

These inequalities represent two lines on a graph. The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap. To visualize this, you'd graph both lines: the first with a solid line (because of the