01 · Explore
Introduction to Inequalities
In mathematics, not every relationship between quantities is one of equality. Inequalities describe situations where one value is less than, greater than, or at most equal to another.
An inequality is formed when two real numbers or algebraic expressions are related by the symbols < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). For example, if your classroom can occupy at most 60 tables or chairs or both, we represent this using the '≤' sign.
Inequalities are classified into different types. Numerical inequalities involve only numbers, such as 3 < 5. Literal inequalities involve variables, such as x > 2. Double inequalities, like 3 < x < 5, describe a variable bounded by two values.
We also distinguish between strict and slack inequalities. Strict inequalities use < or > and do not allow for equality. Slack inequalities use ≤ or ≥, indicating that the quantities could be equal. Linear inequalities in one variable take the form ax + b < 0 (where a ≠ 0), while those in two variables appear as ax + by < c.
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What is the difference between a strict inequality and a slack inequality?
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Rules for Solving Inequalities
Solving an inequality means finding all values of the variable that make the statement true. This process follows specific rules similar to solving equations, with one critical exception.
Rule 1: Equal numbers may be added to or subtracted from both sides of an inequality without changing the inequality sign. For example, if x - 3 < 5, adding 3 to both sides gives x < 8.
Rule 2: Both sides of an inequality can be multiplied or divided by the same positive number without affecting the sign. However, if you multiply or divide both sides by a negative number, the sign of the inequality must be reversed. For example, if -2x < 6, dividing by -2 results in x > -3.
This reversal is necessary because while 3 > 2 is true, multiplying by -1 gives -3 and -2; since -3 is to the left of -2 on the number line, -3 < -2 is the correct relationship.
| Operation | Effect on Inequality Sign | Example |
|---|---|---|
| Addition/Subtraction | No Change | x + 2 < 5 becomes x < 3 |
| Multiplication by Positive | No Change | 2x < 10 becomes x < 5 |
| Multiplication by Negative | Sign Reverses | -2x < 10 becomes x > -5 |
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Why does the inequality sign reverse when multiplying by a negative number?
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03 · Explore
Algebraic Solutions in One Variable
Solving linear inequalities involves isolating the variable using the rules of inequalities. The solution set depends on the domain of the variable, such as natural numbers, integers, or real numbers.
When solving 30x < 200, we divide both sides by 30 to get x < 20/3 (or x < 6.66...). If x is a natural number, the solution set is {1, 2, 3, 4, 5, 6}. If x is an integer, the solutions include zero and negative values, represented as {..., -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}.
For real numbers, the solution is often expressed in interval notation. If x < 2, the solution set is (-∞, 2). If x ≥ 8, the solution set is [8, ∞). The square bracket indicates that the endpoint is included (slack inequality), while the parenthesis indicates it is excluded (strict inequality).
Solving a Linear Inequality
4x + 3 < 6x + 7
Subtract 6x from both sides: 4x - 6x + 3 < 7, which simplifies to -2x + 3 < 7. Subtract 3 from both sides: -2x < 4. Divide by -2 and reverse the sign: x > -2. The solution set is (-2, ∞).
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Solve 5x - 3 < 3x + 1 for real x.
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Graphical Representation on Number Line
Visualizing solutions on a number line helps in understanding the range of values that satisfy an inequality.
To represent a strict inequality like x < a or x > a, we place an open circle at 'a' to show that 'a' itself is not included. We then darken the line to the left for 'less than' or to the right for 'greater than'.
For slack inequalities like x ≤ a or x ≥ a, we use a solid (dark) circle at 'a' to indicate that the endpoint is part of the solution. This distinction is vital for accurately communicating the boundaries of the solution set.
Steps to Graph an Inequality
- 1
Identify Boundary
Find the value 'a' where the inequality starts or ends.
- 2
Choose Circle Type
Use an open circle for < or > and a solid circle for ≤ or ≥.
- 3
Shade the Region
Darken the line to the left of the number 'a' for x < a or x ≤ a, and to the right for x > a or x ≥ a.
The process of translating an algebraic inequality into a visual representation on a one-dimensional number line.
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How would you represent x ≥ 1 on a number line?
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Solving Systems of Linear Inequalities
A system of inequalities consists of two or more inequalities that must be satisfied simultaneously. The solution is the intersection of the individual solution sets.
To solve a system like 3x - 7 < 5 + x and 11 - 5x ≤ 1, we solve each inequality independently. The first yields x < 6 and the second yields x ≥ 2. The combined solution is the set of all x such that 2 ≤ x < 6.
Graphically, this is represented by the overlapping region of the two individual graphs on the number line. In this case, it would be the segment between 2 and 6, with a solid circle at 2 and an open circle at 6.
Simultaneous Solution
-8 ≤ 5x - 3 < 7
Add 3 to all parts: -5 ≤ 5x < 10. Divide all parts by 5: -1 ≤ x < 2. The solution is the interval [-1, 2).
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If a system results in x > 5 and x < 3, what is the solution set?
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06 · Explore
Applications of Linear Inequalities
Inequalities are essential for solving real-world problems involving constraints, averages, and ranges.
In academic grading, inequalities help determine required scores. For example, if a student needs an average of at least 60 marks over three exams and has scored 62 and 48 in the first two, the inequality (62 + 48 + x)/3 ≥ 60 helps find the minimum marks 'x' needed in the final exam.
In scientific experiments and fields like statistics, economics, or psychology, inequalities describe ranges of tolerance. Converting temperatures between Celsius (C) and Fahrenheit (F) using C = 5/9(F - 32) allows us to find the Fahrenheit range for a solution that must be kept between 30°C and 35°C.
Manufacturing and mixture problems also rely on inequalities. If a manufacturer wants a final acid solution to be between 15% and 18% concentration, they use inequalities to calculate the exact volume of a specific concentration to add to an existing stock.
Temperature Range Calculation
30 < 5/9(F - 32) < 35
Multiply by 9/5: 54 < F - 32 < 63. Add 32 to all parts: 86 < F < 95. The temperature must be between 86°F and 95°F.
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A student has marks 87, 92, 94, and 95. What minimum marks in the 5th exam are needed for an average of at least 90?
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