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A short class with Sarah and her students. The other ways are under More.
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Today, we're going to practice solving linear inequalities. Who can remind me the general rule when we solve an inequality involving a negative number?
You have to flip the inequality sign when you multiply or divide by a negative!
Exactly! Let's try solving the inequality 3x - 7 > 5. How would you start?
First, add 7 to both sides, so it becomes 3x > 12.
Great! And what comes next?
Then, divide by 3, so x > 4.
Well done! Now, how would we graph this on a number line?
We would use an open circle at 4 and shade to the right.
That's correct! Just remember, open circles mean it's not included. Let's summarize: solving inequalities requires careful tracking of the sign, and graphing them accurately shows the range of possible solutions.
Now that we've solved one-variable inequalities, let's graph some inequalities involving two variables. Who can explain the first step in graphing an inequality like y ≤ 2x + 3?
First, we need to graph the line y = 2x + 3.
Do we use a solid or dashed line?
Excellent question! Since we have 'less than or equal to,' we use a solid line. What comes next?
Then we shade below the line!
Right! Why do we shade below?
Because we are looking for y values that are less than or equal to 2x + 3.
Perfect! So if we test a point like (0,0) and it's true, we shade towards it. Just remember: always check a point to ensure you shade the correct area.
Let's say we have a system of inequalities: x + y ≤ 6 and x ≥ 2. How would we find the solution region?
We can graph both inequalities on the same coordinate plane.
Absolutely! What do we do after graphing them?
We find the overlapping shaded area, which represents the solution to the system.
So if one inequality is a line going up and the other is a vertical line at x = 2, we look for where both shaded areas intersect.
Correct! This intersection is where all conditions are satisfied. Remember, systems can represent real-life constraints that often require simultaneous solutions!
Let’s apply what we learned to a real-world problem: A student has $20 to spend on snacks that cost $2 each. How can we model this situation with an inequality?
We can let x be the number of snacks and write the inequality 2x ≤ 20.
Exactly! And how would we solve that inequality?
We divide both sides by 2, so x ≤ 10. The student can buy up to 10 snacks!
Well done! Always remember to translate everyday situations into inequalities. This helps in making informed decisions within constraints.
An expression that uses inequality signs (<, >, ≤, ≥) instead of an equals sign.
The line in a graph that represents the limits of solutions for two-variable inequalities.
The representation of solutions in graph form, showing the area that satisfies the inequality.
Specific coordinates used to determine which side of a boundary line to shade.
If you see the sign, don't forget the flip, for negative numbers, it’s a little trip.
Imagine a student named Alex, who wants to decide how many books to buy. He has $20 and each book costs $4, leading him to write 4x ≤ 20, deciding how many he can get!
When graphing, think of the phrase 'Line to shine' for where to put the solid line, and 'Dashed for the bash' when it doesn't include.
G.R.A.F. - Graphing Rules: Remember the line type, Right shading, After the test point, Follow the direction.
Term
What is a linear inequality?
Click to flip
Term
What is the meaning of 'shading' in a graph of an inequality?
Click to flip
An inequality that involves a linear function, representing a range of values.
The line that represents the limit of the solution region in a two-variable inequality.
The area of the graph that represents all solutions to the inequality.
A point used to determine which side of the boundary line to shade.
A set of two or more inequalities that are considered simultaneously.
This section contains practice exercises for understanding and applying linear inequalities.
The practice exercises consist of various problems designed to reinforce key concepts related to linear inequalities, including solving inequalities in one and two variables and representing their solutions graphically.
This section includes a variety of practice problems that help you apply and reinforce your understanding of linear inequalities. Each exercise is crafted to cover different aspects of what you've learned in previous sections, such as solving linear inequalities in one variable, graphing inequalities in two variables, and understanding systems of inequalities. These exercises will allow you to practice converting real-world problems into mathematical inequalities and finding solutions using both algebraic and graphical methods. Completing these exercises will enhance your skills and prepare you for real-life applications.