Literal inequalities are mathematical statements that express an order relationship between two expressions, and critically, they involve variables. They use symbols to show if one expression is less than, greater than, less than or equal to, greater than or equal to, or not equal to another expression. Essentially, they are inequalities containing variables, making them solvable for those variables.
Here's a breakdown:
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Definition: A literal inequality is a mathematical sentence comparing two expressions that include at least one variable, using inequality symbols.
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Symbols Used: The common symbols used to represent the relationship between the expressions are:
- < (less than)
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(greater than)
- ≤ (less than or equal to)
- ≥ (greater than or equal to)
- ≠ (not equal to)
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Examples:
x + 3 < 7(x plus 3 is less than 7)2y - 5 ≥ 1(2y minus 5 is greater than or equal to 1)a + b > c(a plus b is greater than c)3z ≠ 9(3z is not equal to 9)
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Solving Literal Inequalities: Similar to solving equations, the goal is to isolate the variable on one side of the inequality. However, there's a crucial rule to remember: when multiplying or dividing both sides of an inequality by a negative number, you must reverse the inequality sign.
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Example of Solving:
Let's solve the inequality-2x < 6.- Divide both sides by -2:
(-2x) / -2 > 6 / -2(Notice the inequality sign is reversed!) - Simplify:
x > -3
- Divide both sides by -2:
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Graphing Literal Inequalities: The solution to a literal inequality can be represented on a number line. Open circles are used for
<and>, while closed circles are used for≤and≥.
In essence, literal inequalities are a fundamental concept in algebra, used to describe and solve a range of problems where exact equality isn't required, but rather a range of possible values.
