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Inequality Calculator

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Example

3x - 5 > -2(x - 10)

Knowledge About Inequality

1.
What is an inequality?

Imagine you have two numbers or expressions. An inequality tells you how they relate to each other, but it's not just a simple 'equals' sign (=). Instead, you might see symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Think of it as a teeter-totter at the playground: one side is always either higher or lower than the other, right? For example, x > 3 tells us that x is any number greater than 3.
2.
How to graph inequalities?
Graphing inequalities is a fun way to see all the possible solutions at once! Here’s how you can do it:
1. Draw the number line or coordinate plane: Depending on whether it’s one variable (like x > 3) or two variables (like y > 2x + 1).
2. Plot the critical points: For x > 3, you’d put a hollow dot on 3 if it’s just greater than (not including 3), or a solid dot if it’s greater than or equal to (including 3).
3. Shade the region: For x > 3, shade everything to the right of 3. For something like y > 2x + 1, you’d shade above the line.
3.
How to solve inequalities?
Solving an inequality is like finding all the numbers that make the statement true. You can use your algebra skills (addition, subtraction, multiplication, division) to manipulate the inequality and isolate the variable on one side:
1. Isolate the variable: Just like with equations, get your variable x on one side by itself. If you have 3x + 2 > 11, subtract 2 from both sides to get 3x > 9.
2. Divide or multiply to solve: Continuing from above, divide both sides by 3 to get x > 3.
3. Remember to flip the inequality: If you ever multiply or divide by a negative number, remember to flip the inequality sign.
4.
How to solve absolute value inequalities?
Absolute value inequalities involve expressions wrapped in those fancy double bars (| |). They basically ask you to find the solutions where the distance between the expression and zero is less than, greater than, or equal to a certain number, and they are solved in a unique way:
1. Write two inequalities: If you have |x - 2| < 5, think of it as two inequalities: x - 2 < 5 and x - 2 > -5.
2. Solve both: Solve these like normal inequalities to find x < 7 and x > -3, which tells you x is between -3 and 7.
5.
Real-world Applications of Inequalities
Inequalities pop up in daily life more than you might think:
- Budgeting: In budgeting, inequalities are used to make sure that expenses do not exceed income. The expression \text{balance} - \text{price} \geq 0 checks whether you can afford a purchase without going into debt, which is a practical application of inequalities in financial management.
- Cooking: Setting an oven within a certain temperature range, such as 350^\circ \leq \text{oven temperature} \leq 400^\circ, ensures that food cooks properly without burning or undercooking. This is a direct application of inequalities to maintain control over cooking conditions.
- Sports: In sports, inequalities are used to denote what a team must score or how many points a team should accumulate to advance in a tournament, for instance. Such cases may involve quite complicated inequalities when several conditions or tie-breakers are considered.
6.
Fun Facts about Inequalities
- Ancient Concepts: Although Diophantus and others, including Thomas Harriot, were concerned with ideas having to do with inequalities, the formalization and systematic study of inequalities came much later. For example, Thomas Harriot did work involving notation and solving inequalities, which provided a background to future developments.
- Language and Math: In mathematics, the word 'inequality' does mean just that: 'not equal', and the word describes relationships in which one amount is either less than, greater than, or not equal to another. This word captures a basic concept in mathematics that goes beyond simple equations.
- Puzzles and Problems: Logic puzzles and brain teasers frequently impose the exact conditions and constraints under which solvers are challenged to use judgment and strategy in terms of the relationships that the variables have between them.
Inequalities help us describe a whole range of possibilities instead of one fixed answer, making them super useful in scenarios where flexibility and ranges are involved. Cool, right?

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Inequality Calculator

Example

Knowledge About Inequality

What is an inequality?

How to graph inequalities?

How to solve inequalities?

How to solve absolute value inequalities?

Real-world Applications of Inequalities

Fun Facts about Inequalities

Still have questions? Ask UpStudy online

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