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Interactive Audio Lesson
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Solving Linear Inequalities
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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.
Graphing Inequalities in Two Variables
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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.
Understanding Systems of Inequalities
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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!
Real-World Applications
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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.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
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.
Detailed
Practice Exercises
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.
Audio Book
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Exercise 1: Solving and Graphing Inequalities
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- Solve and graph: 3x - 7 > 5
Detailed Explanation
To solve the inequality 3x - 7 > 5, we first isolate the variable x. We can do this by adding 7 to both sides: 3x > 5 + 7, which simplifies to 3x > 12. Next, we divide both sides by 3 to solve for x: x > 4. Now, we need to graph this inequality on a number line. Since it is a 'greater than' inequality, we use an open circle at 4 and shade to the right to indicate all numbers greater than 4.
Examples & Analogies
Imagine you are planning a party and want to invite more than 4 friends. Since you can invite 5 or more friends, the number of friends you can invite is any number greater than 4, represented on a number line where you mark 4 and shade everything to the right.
Exercise 2: Graphing Inequalities
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- Graph: 2x + y ≤ 6
Detailed Explanation
To graph the inequality 2x + y ≤ 6, we first rewrite the inequality in slope-intercept form, which is y ≤ -2x + 6. The boundary line is y = -2x + 6, which we graph using a solid line because it includes the equality (≤). Next, we choose a test point (0, 0) to see where to shade. Since 2(0) + 0 ≤ 6 is true, we shade below the line. This represents all points (x, y) that satisfy the inequality.
Examples & Analogies
Think of a budget constraint where you can spend a maximum of $6 on snacks, where 2x is the amount spent on one type of snack and y is the amount spent on another. Graphing this inequality helps represent all the combinations of snacks you can buy without exceeding your budget.
Exercise 3: Solving Inequalities
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- Solve: -2(x - 4) ≥ 10
Detailed Explanation
To solve -2(x - 4) ≥ 10, we begin by distributing the -2: -2x + 8 ≥ 10. Next, we isolate the variable by subtracting 8 from both sides, yielding -2x ≥ 2. Then, we divide both sides by -2, remembering to reverse the inequality sign, which gives us x ≤ -1. This inequality shows that x can take any value up to and including -1.
Examples & Analogies
Imagine a situation where you are required to score at least a certain number of points in a game to win. Here, getting a score less than or equal to -1 might mean you have certain limitations, like needing to clear -1 levels in a video game scenario.
Exercise 4: Creating an Inequality
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- A ticket costs $12. Write an inequality to find how many tickets a person can buy with $100.
Detailed Explanation
To find out how many tickets (let's call the number of tickets x) a person can buy for $100 when each ticket costs $12, we set up the inequality 12x ≤ 100. This means the total cost (12 times the number of tickets) cannot exceed $100. To find x, we divide both sides by 12, leading to x ≤ 8.33. Since a person cannot buy a fraction of a ticket, this means they can buy at most 8 tickets.
Examples & Analogies
Consider planning a movie night where each ticket costs money. Keeping track of how many you can get while sticking to your budget helps you organize your finances better. You can buy 8 tickets but not more, emphasizing the importance of budget management.
Exercise 5: Shading the Solution of a System
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- Shade the solution of the system:
o x + y < 5
o x > 1
Detailed Explanation
To find the solution to the system of inequalities, we first graph each inequality separately. For x + y < 5, we rearrange it to y < -x + 5, and draw the line y = -x + 5 using a dashed line (since it is 'less than'). Then we test a point to see where to shade. For x > 1, we draw a dashed vertical line at x = 1 and shade to the right. The solution is the overlapping shaded region that satisfies both inequalities.
Examples & Analogies
Think of a scenario where two friends want to plan a picnic. One wants to keep the total number of guests under 5, while another can only invite guests starting from just over 1. The area that satisfies both conditions helps them find how many friends they can actually invite together.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Linear Inequality: An expression that uses inequality signs (<, >, ≤, ≥) instead of an equals sign.
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Boundary Line: The line in a graph that represents the limits of solutions for two-variable inequalities.
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Shading: The representation of solutions in graph form, showing the area that satisfies the inequality.
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Test Points: Specific coordinates used to determine which side of a boundary line to shade.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example 1: Solve and graph the inequality 5x - 9 < 6. Solution: x < 3.
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Example 2: Graph the inequality x + y ≥ 4. The boundary line is solid, and we shade above it.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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If you see the sign, don't forget the flip, for negative numbers, it’s a little trip.
📖 Fascinating Stories
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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!
🧠 Other Memory Gems
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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.
🎯 Super Acronyms
G.R.A.F. - Graphing Rules
- Remember the line type
- Right shading
- After the test point
- Follow the direction.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Linear Inequality
Definition:
An inequality that involves a linear function, representing a range of values.
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Term: Boundary Line
Definition:
The line that represents the limit of the solution region in a two-variable inequality.
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Term: Shaded Region
Definition:
The area of the graph that represents all solutions to the inequality.
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Term: Test Point
Definition:
A point used to determine which side of the boundary line to shade.
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Term: System of Inequalities
Definition:
A set of two or more inequalities that are considered simultaneously.