Introduction to Linear Inequations

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Questions and Answers

What is the meaning of the inequality sign $\geq$ ?

Greater than or equal to

What is a linear inequality?

An inequality that involves a linear expression.

What is the difference between a linear equation and a linear inequality?

A linear equation represents equality, while a linear inequality represents an unequal relationship.

What is the purpose of solving a linear inequality?

<p>To find the values of the variable that satisfy the inequality.</p> Signup and view all the answers

The solution to a linear inequality is always a single value.

<p>False (B)</p> Signup and view all the answers

What are the two basic rules for solving a linear inequality?

<p>Rule 1: Adding or subtracting the same term on both sides of the inequality does not change the solution. Rule 2: Multiplying or dividing both sides of the inequality by the same positive number does not change the solution. However, multiplying or dividing by a negative number requires flipping the inequality sign.</p> Signup and view all the answers

Flashcards

Inequation

Mathematical statements comparing two expressions using inequality signs like >, <, ≥, ≤.

Linear Inequation in One Variable

An inequation with one variable where the highest power of the variable is 1.

Linear Expression

A mathematical expression that involves a variable raised to the power of 1 ('x').

Solution of an Inequation

A set of values that satisfy the inequation.

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Solving an Inequation

To find the solution set, transform the inequation into an equivalent inequation where the variable is isolated on one side.

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Addition or Subtraction Property of Inequality

Adding or subtracting the same number to both sides of an inequation does not change the solution set.

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Multiplication or Division Property of Inequality (Positive Number)

Multiplying or dividing both sides of an inequation by the same positive number does not change the solution set.

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Multiplication or Division Property of Inequality (Negative Number)

Multiplying or dividing both sides of an inequation by the same negative number reverses the direction of the inequality sign.

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Graphical Representation of the Solution Set

The solution set can be represented on a number line.

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Reversing the Inequality Sign

When solving an inequation, if you multiply or divide both sides by a negative number, you must reverse the inequality sign.

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Equivalent Inequations

Transforming an equation into an equivalent equation without changing the solution set.

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Transformations of Inequations

Operations that produce equivalent inequations.

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Universal Solution Set

A solution set that includes all real numbers.

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Empty Solution Set

A solution set that includes no real numbers.

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Finite and Infinite Solution Sets

The solution set of a linear inequation can either be a finite set or an infinite set.

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Graphical Representation on a Number Line

To represent the solution set graphically, you can use a number line and shade the area that corresponds to the solutions.

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Closed and Open Intervals

An inequation can have a solution that is a closed interval or an open interval depending on whether the inequality sign includes equality.

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Solution Set in a Specific Domain

When solving an inequation, the solution set may be restricted to a specific domain.

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Representations of Solution Sets

The solution set of an inequation can be expressed in different forms like interval notation, set builder notation, or graphical representation.

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Study Notes

Introduction to Linear Inequations

  • Linear inequation: A statement that compares two expressions using inequality symbols (>, <, ≥, ≤).
  • Variables: Quantities that can take on different numerical values.
  • Conditions: Possible relations between quantities (x > y, x ≥ y, x < y, x ≤ y).
  • Examples: x < 8, x ≥ 5, x + 4 ≤ 3.

Linear Inequations in One Variable

  • Real numbers: Numbers that can be represented on a number line, including integers, fractions, and decimals.
  • Forms: ax + b > c, ax + b < c, ax + b ≥ c, ax + b ≤ c.
  • Interpretation: Example ax + b > c means 'ax + b is greater than c'.
  • Signs of inequality: >, <, ≥, ≤.

Solving Linear Inequations Algebraically

  • Solving: Finding the variable values satisfying the inequality.
  • Rule 1: Transposing a positive term changes its sign to negative when moving to the other side of the inequality. (e.g., 2x + 3 > 7 becomes 2x > 7 - 3)
  • Rule 2: Transposing a negative term changes its sign to positive when moving to the other side of the inequality. (e.g., 2x – 3 > 7 becomes 2x > 7 + 3)

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