## Part 1: Introduction to Inequalities

Welcome to MKSprep’s SAT Preparation Course! This comprehensive course is designed to guide you through all the necessary SAT topics to maximize your potential and score. Our preparation center, located in the heart of Putalisadak, Kathmandu, Nepal, is dedicated to supporting students in their journey towards SAT success.

Our next topic is Inequalities. Inequalities are a crucial component of the SAT Mathematics section, and understanding them can significantly boost your score. This topic can seem daunting initially, but fear not! We’re here to break it down for you in an easy-to-understand way.

### What are Inequalities?

In the realm of mathematics, an inequality is a relation between two expressions that may not be equal. It is represented using inequality symbols: less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥).

For example, the inequality x > 3 means that x is any number greater than 3. Similarly, y ≤ 5 means y is any number less than or equal to 5.

### Solving Inequalities

Solving inequalities involves finding all possible values of the variable that make the inequality true. The methods for solving inequalities are very similar to those for solving equations, with the primary difference being the use of inequality symbols instead of equality symbols.

For instance, to solve the inequality 2x + 3 < 7, you would first subtract 3 from both sides to get 2x < 4, then divide both sides by 2 to find that x < 2.

### Importance in the SAT

The SAT often tests your understanding of inequalities in various ways. You might have to solve inequalities, interpret them in the context of a word problem, or graph them on a number line. Understanding inequalities can help you tackle these problems confidently and effectively.

We will be taking you through the ins and outs of inequalities in the upcoming sections, including solving and graphing inequalities, applying them in word problems, and exploring compound and absolute value inequalities.

Stay tuned for our next module, where we will dive into the process of solving linear inequalities in more detail.

[Next: Solving Linear Inequalities]

## Part 2: Solving Linear Inequalities

Welcome back to MKSprep’s SAT prep course! We are located in Putalisadak, Kathmandu, Nepal, and we’re here to help you conquer the SAT. In this module, we’re going to delve into solving linear inequalities.

Linear inequalities are similar to linear equations, but instead of an equals sign (=), they use inequality signs (<, ≤, >, ≥). Solving them involves finding the range of values that make the inequality statement true.

### How to Solve Linear Inequalities

The process of solving linear inequalities is quite similar to that of solving linear equations. To isolate the variable, you perform the same operations (addition, subtraction, multiplication, and division) on both sides of the inequality. The crucial thing to remember is that if you multiply or divide by a negative number, the inequality sign must be flipped.

For example, consider the inequality -2x + 3 > 1:

1. Subtract 3 from both sides: -2x > -2.
2. Divide by -2 (remember to flip the inequality sign): x < 1.

Therefore, the solution to the inequality -2x + 3 > 1 is x < 1.

### Graphing Linear Inequalities

When you solve a linear inequality, you can represent the solution on a number line. For x < 1, you’d mark a point above 1 on the number line and then draw a line extending to the left from this point. Since 1 is not included in the solution (because the inequality is “less than” and not “less than or equal to”), you would use an open circle at 1.

### Practice

Try solving these linear inequalities:

1. 4x – 7 ≤ 5
2. -3x + 2 > 8
3. 5 – 2x < 15

In the next module, we’ll delve into how to solve and graph compound inequalities. These problems involve more than one inequality and are a little trickier, but with our guidance, you’ll master them in no time!

[Next: Compound Inequalities]

## Part 3: Compound Inequalities

Great to have you back in our SAT prep course at MKSprep! We’re located in the vibrant locale of Putalisadak, Kathmandu, Nepal. Today, we’re moving forward in our exploration of inequalities and focusing on a more complex type – compound inequalities.

### What are Compound Inequalities?

Compound inequalities are statements with two inequalities joined by the words ‘and’ or ‘or.’ Here’s what each of these types means:

1. ‘And’ Inequalities: When two inequalities are connected by ‘and,’ it means that both conditions must be satisfied simultaneously. In the compound inequality 1 < x and x < 3, x must be greater than 1 and less than 3. We often write this compound inequality as 1 < x < 3.
2. ‘Or’ Inequalities: When two inequalities are connected by ‘or,’ it means that either condition (or both) must be satisfied. In the compound inequality x < -2 or x > 3, x must be less than -2 or greater than 3.

### Solving and Graphing Compound Inequalities

Solving compound inequalities is similar to solving simple inequalities, but you have to solve for two conditions instead of one. Graphing them can provide a visual representation of the solutions.

1. ‘And’ Inequalities: For the inequality 1 < x < 3, on a number line, you’d draw an open circle at 1 and 3 and a line connecting them. This shows all the numbers between 1 and 3 are solutions.
2. ‘Or’ Inequalities: For the inequality x < -2 or x > 3, you’d draw open circles at -2 and 3. You’d then draw a line extending to the left from -2 and to the right from 3. This shows all the numbers less than -2 and greater than 3 are solutions.

### Practice

Try solving and graphing these compound inequalities:

1. -3 ≤ x and x ≤ 4
2. x < -1 or x > 3
3. -2 < x < 2

Compound inequalities can seem challenging at first, but once you’ve grasped the concept, they’re just as straightforward as simple inequalities. In our next section, we’ll apply our knowledge of inequalities to real-world situations by solving word problems.

[Next: Inequality Word Problems]

## Part 4: Inequality Word Problems

Welcome back to MKSprep’s SAT prep course, located in the bustling hub of Putalisadak, Kathmandu, Nepal. As we journey through the topic of inequalities, we now arrive at an integral part of the SAT math section: word problems involving inequalities.

### What are Inequality Word Problems?

Inequality word problems, just like regular word problems, provide a real-world scenario, but the question leads to an inequality instead of an equation. These problems can involve topics such as money, time, distance, or any other real-life scenario that can be represented mathematically.

### How to Solve Inequality Word Problems

Solving inequality word problems involves several steps:

1. Understand the problem: Read it carefully and determine what it asks.
2. Define the variables: Assign a variable to the unknown quantity.
3. Set up the inequality: Translate the words into a mathematical inequality.
4. Solve the inequality: Use the methods we’ve discussed to solve the inequality.
5. Interpret the solution: Make sure your solution makes sense in the context of the problem.

### Example

Let’s go through an example to illustrate the process:

You have \$500 to spend on books and pens. Books cost \$30 each, and pens cost \$10 each. If you want to buy at least 10 books, how many pens can you buy?

1. Understand the problem: You want to buy at least 10 books and as many pens as possible with \$500.
2. Define the variables: Let’s define B as the number of books and P as the number of pens.
3. Set up the inequality: The total cost should be less than or equal to \$500. So, \$30B + \$10P ≤ \$500.
4. Solve the inequality: If B is at least 10, replace B with 10 in the inequality: \$30(10) + \$10P ≤ \$500. This simplifies to \$300 + \$10P ≤ \$500, and then to \$10P ≤ \$200. Dividing by 10, we get P ≤ 20.
5. Interpret the solution: You can buy up to 20 pens.

### Practice

Try solving these inequality word problems:

1. You’re babysitting to earn money to buy a \$50 concert ticket. If you earn \$8 per hour, how many hours do you need to work?
2. A zoo requires at least 100 lbs of food each day for its animals. If a bag of food costs \$5 and weighs 2 lbs, how much will the zoo spend on food in a week?

Next, we’ll expand our understanding of inequalities to include absolute value inequalities, a topic that can come up on the SAT and further enriches our understanding of mathematics.

[Next: Absolute Value Inequalities]

## Part 5: Absolute Value Inequalities

Welcome back to MKSprep’s SAT preparation course, located in the thriving heart of Putalisadak, Kathmandu, Nepal. We’re continuing our exploration of inequalities, and in this module, we’re introducing a new concept: absolute value inequalities.

### What are Absolute Value Inequalities?

Absolute value inequalities are inequalities that contain absolute value expressions. An absolute value measures distance from zero, so |x| represents the distance of x from zero on a number line. It’s always positive or zero, never negative.

### Solving Absolute Value Inequalities

Solving absolute value inequalities involves splitting the inequality into two separate inequalities, depending on whether the inequality sign is ‘less than’ (< or ≤) or ‘greater than’ (> or ≥).

1. ‘Less Than’ Inequalities (< or ≤): If |x| < a, then -a < x < a. This means x is between -a and a.
2. ‘Greater Than’ Inequalities (> or ≥): If |x| > a, then x < -a or x > a. This means x is either less than -a or greater than a.

### Example

Consider the inequality |x – 3| ≤ 5. Because this is a ‘less than inequality, we split it into two:

-5 ≤ x – 3 ≤ 5

Solving this compound inequality gives us the solution:

-2 ≤ x ≤ 8

This means any number between -2 and 8 (inclusive) would satisfy the original absolute value inequality.

### Practice

Try solving these absolute value inequalities:

1. |x + 2| < 7
2. |3x – 1| ≥ 6

Remember, mastering absolute value inequalities requires practice. It might seem tricky initially, but soon it will become second nature.

Next up, we’ll apply our understanding of inequalities to an important SAT math topic: systems of inequalities. See you there!

[Next: Systems of Inequalities]

## Part 6: Systems of Inequalities

Welcome back to MKSprep’s SAT preparation course. We are located in Putalisadak, Kathmandu, Nepal, and are devoted to guiding you on your journey towards SAT success. In this section, we’ll venture into a more complex area of inequalities: systems of inequalities.

### What are Systems of Inequalities?

Just as a system of equations is a set of two or more equations, a system of inequalities is a set of two or more inequalities. In a system of inequalities, you’re looking for the set of values that satisfy all the inequalities in the system.

### Solving Systems of Inequalities

Solving systems of inequalities is primarily a graphical procedure:

1. Graph each inequality: Start by graphing each on the same axes. The solution to each inequality is the area shaded by that inequality.
2. Find the overlap: The solution to the system of inequalities is the overlap area that satisfies all the inequalities.
3. Check the boundaries: If the inequality includes an equals sign (≥ or ≤), the line itself is included in the solution and is drawn as a solid line. If not (> or <), it’s represented as a dashed line.

### Example

Consider the system of inequalities:

1. y ≤ x – 2
2. y > -x

First, graph both inequalities on the same set of axes. Then, identify the area where the shading from both inequalities overlaps. This overlapping area represents all the (x, y) pairs that satisfy both inequalities.

### Practice

Try solving these systems of inequalities:

1. y ≥ 2x and y < x + 3
2. y ≤ -x + 4 and y > x – 1

Mastering systems of inequalities takes practice, but it’s a valuable skill for the SAT and beyond. Next, we’ll apply our knowledge of inequalities to quadratic inequalities, which is our final topic in this SAT prep course on inequalities.

Greetings once again from MKSprep’s SAT preparation course in the heart of Putalisadak, Kathmandu, Nepal. We’re now entering the domain of quadratic inequalities, an important topic for the SAT.

Quadratic inequalities are inequalities that can be written in the form ax² + bx + c > 0 or ax² + bx + c < 0, where a, b, and c are constants. They are like quadratic equations but have an inequality sign instead of an equals sign.

Solving quadratic inequalities involves several steps:

1. Solve the related quadratic equation: First, find the roots of the related quadratic equation (replace the inequality sign with an equals sign).
2. Use the roots to define intervals on the number line: The roots of the quadratic equation divide the number line into intervals.
3. Test the sign of the quadratic expression in each interval: Pick a number from each interval and substitute it into the quadratic inequality. If the inequality is satisfied, that interval is part of the solution.
4. Combine the solution intervals: Write the solution as an inequality or a union of intervals.

### Example

Let’s solve the quadratic inequality x² – 4x + 3 > 0.

1. Solve the related quadratic equation: The roots of x² – 4x + 3 = 0 are x = 1 and x = 3.
2. Use the roots to define intervals on the number line: The roots split the number line into three intervals: (-∞, 1), (1, 3), and (3 ∞).
3. Test the sign of the quadratic expression in each interval: Choose x = 0 (from the first interval), x = 2 (from the second interval), and x = 4 (from the third interval). Substitute these values into the original inequality. The first and third intervals satisfy the inequality.
4. Combine the solution intervals: The solution is x < 1 or x > 3.

### Practice

1. x² – 5x + 6 ≥ 0
2. 2x² – 3x – 2 < 0

Understanding quadratic inequalities is a crucial skill for the SAT. Stay tuned for the final section of our inequality study, where we’ll comprehensively review all the topics covered so far.

[Next: Comprehensive Review]

## Part 8: Comprehensive Review

We’re delighted to have you back at MKSprep’s SAT preparation course in Putalisadak, Kathmandu, Nepal. This is the final part of our study on inequalities, where we’ll review all the concepts we’ve covered so far and integrate them through a set of practice problems.

### Review

Throughout this course, we’ve covered:

1. Simple Inequalities: Inequalities with only one variable, which we solve like equations, but remember the rule of flipping the inequality sign when multiplying or dividing by a negative number.
2. Compound Inequalities: Inequalities connected by ‘and’ or ‘or,’ where we solve for two conditions simultaneously.
3. Inequality Word Problems: Real-world scenarios leading to an inequality.
4. Absolute Value Inequalities: Inequalities that contain absolute value expressions, where we split the inequality into two separate inequalities depending on the inequality sign.
5. Systems of Inequalities: A set of two or more inequalities, where we graph each inequality and find the overlapping area that satisfies all the inequalities.
6. Quadratic Inequalities: Inequalities that can be written in a quadratic form, where we solve the related quadratic equation and use the roots to test the sign of the quadratic expression in each interval.

### Comprehensive Practice

Now, let’s test your knowledge of inequalities with these practice problems:

1. Solve and graph the inequality: -3x + 2 > 1.
2. Solve the compound inequality: -2 ≤ 3x – 1 < 5.
3. Solve the word problem: You’re saving money for a \$200 gadget. If you save \$15 per week, how many weeks do you need to save for?
4. Solve the absolute value inequality: |2x – 5| ≤ 3.
5. Graph the system of inequalities: y ≥ x and y < 2x + 3.
6. Solve the quadratic inequality: x² – x – 6 > 0.

### Conclusion

Understanding inequalities is essential to SAT preparation, providing the foundation for many other topics and problems you will encounter in the math section. Keep practicing and reinforcing these concepts until they become second nature. With diligent work, you’ll be well on your way to achieving your SAT goals!

Thanks for joining us in this in-depth study on inequalities. Stay tuned for our next SAT prep course!