How to Solve Inequalities in Slope-Intercept Form
Quick Answer
To solve the inequality 3x − y > −16 in slope-intercept form, isolate y to get y < 3x + 16. Remember, multiplying by a negative flips the inequality sign.
Understanding how to solve inequalities in slope-intercept form is a crucial skill in algebra. Let's break down the process using the inequality 3x − y > −16 as our example.
### Step 1: Start with the Original Inequality
We begin with the inequality:
3x − y > −16
Our goal is to isolate y on one side of the inequality. This will help us express the inequality in the slope-intercept form, which is typically written as y = mx + b, where m is the slope and b is the y-intercept.
### Step 2: Move 3x Away from y
To isolate y, we need to move the 3x term to the other side. We can do this by subtracting 3x from both sides of the inequality:
3x − y − 3x > −16 − 3x
This simplifies to:
−y > −16 − 3x
### Step 3: Isolate y
Now we have −y on the left side. To get y by itself, we must multiply both sides by -1. However, it’s important to remember that when we multiply or divide an inequality by a negative number, we must flip the inequality sign. So:
−1(−y) < −1(−16 − 3x)
This simplifies to:
y < 16 + 3x
### Step 4: Finalize the Slope-Intercept Form
Now, we can rearrange the equation to write it in the typical slope-intercept format. We usually write it as:
y < 3x + 16
This tells us that the slope (m) is 3 and the y-intercept (b) is 16. It also indicates that for any x-value, the corresponding y-value must be less than the expression 3x + 16.
### Why Did the Inequality Sign Change?
You might wonder why the inequality sign flipped when we multiplied by -1. This is a key rule in algebra: multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. For example, if you have -2 < 3 and you multiply both sides by -1, you get 2 > -3.
### Real-World Applications
Understanding how to manipulate inequalities is important in various fields, including economics, physics, and everyday problem-solving. For instance, if you are budgeting and want to ensure your expenses stay below a certain amount, setting up inequalities can help you make informed decisions.
By practicing these steps and understanding the reasoning behind them, you will become more confident in working with inequalities in slope-intercept form. Remember, practice makes perfect, so try solving more inequalities to reinforce your skills!
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