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Algebraic Inequalities

Algebraic inequalities compare values using symbols to express relationships such as greater than or less than. This overview covers their properties, such as the addition, subtraction, multiplication, and division properties, and the transitive and comparison properties. It also explains how to solve linear and quadratic inequalities, including the use of number lines and graphical representation to visualize solutions. Understanding these concepts is crucial for mastering algebra.

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1

Inequality Symbols Meaning

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'>': greater than, '<': less than, '≥': greater than/equal to, '≤': less than/equal to.

2

Inequality Solution Set

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The solution to an inequality is a range of values satisfying the comparison, not just one value.

3

Inequality Example Interpretation

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For 'x + 1 > 3', x must be any value greater than 2 to satisfy the inequality.

4

Performing the same mathematical operation on both sides of an ______ does not change its direction, unless you ______ or ______ by a negative number.

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inequality multiply divide

5

According to the ______ property, if one value is greater than a second, and the second is greater than a third, then the first value is greater than the ______.

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transitive third

6

Example of linear inequality

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x + 2 < 7; variable to the first power, simpler to solve.

7

Example of quadratic inequality

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x^2 + x - 20 < 0; variable to the second power, requires complex algebra.

8

In the process of solving ______ inequalities, the goal is to get the variable alone on one side.

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linear

9

Values not part of the solution set in inequalities are shown with ______ circles on a number line.

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open

10

Critical values in quadratic inequalities

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Found by solving the quadratic equation set to zero; divide the number line into intervals.

11

Sign determination in intervals

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Check the sign of the quadratic expression within each interval to determine inclusion in solution set.

12

Graphical method for inequalities

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Visualize solutions by examining where the graph lies relative to the x-axis; above or below depends on inequality.

13

When illustrating inequalities, the solution is the area where ______ is less than ______, indicated by shading.

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f(x) g(x)

14

Inequality Solution Representation

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Solutions expressed on number line or graphically.

15

Inequality Solution Spectrum

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Inequalities yield range of solutions, not just one.

16

Inequality Types Mastery

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Understanding both linear and quadratic inequalities is crucial.

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Understanding Algebraic Inequalities

Algebraic inequalities are expressions that compare two values, using symbols to indicate whether one value is greater than, less than, or not equal to another. These symbols are ">" (greater than), "
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Properties of Inequalities

Inequalities adhere to several properties that allow for the manipulation of their terms while preserving their validity. These properties include the addition, subtraction, multiplication, and division properties, which state that performing the same operation on both sides of an inequality does not affect its direction, except when multiplying or dividing by a negative number, which reverses the inequality. The transitive property allows us to conclude that if a > b and b > c, then a > c. The comparison property asserts that adding the same positive number to both sides of an inequality results in a greater sum on the side where the number was larger initially.

Types of Inequalities

Inequalities are classified by the degree of the variable involved. Linear inequalities involve variables to the first power and are represented by expressions such as x + 2 < 7. Quadratic inequalities include variables to the second power, exemplified by x^2 + x - 20 < 0. The methods for solving these inequalities differ, with linear inequalities typically being more straightforward to solve than quadratic inequalities, which may require more complex algebraic techniques.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side of the inequality. The solution set consists of all real numbers that make the inequality true. Open circles on a number line represent values that are not included in the solution set (indicated by ">" or "

Solving Quadratic Inequalities

Solving quadratic inequalities requires setting the inequality to zero on one side and finding the critical values by solving the corresponding quadratic equation. These critical values divide the number line into intervals. The sign of the quadratic expression within each interval determines whether the interval is part of the solution set. The solution to the inequality is the set of intervals where the quadratic expression satisfies the inequality, and this can be visualized by examining the regions where the graph of the quadratic function lies above or below the x-axis, depending on the inequality's direction.

Graphical Representation of Inequalities

Graphical representation of inequalities involves sketching the functions and shading the regions where the inequality is true. For instance, if f(x) < g(x), the area where the graph of f(x) is below g(x) represents the solution. The points of intersection, or critical values, define the boundaries of these regions. Dotted lines are used to represent strict inequalities ("

Key Takeaways on Inequalities

Inequalities are an essential component of algebra, offering a spectrum of solutions rather than a single answer. They can be manipulated using specific properties, and their solutions can be expressed on a number line or through graphical representation. Mastery of solving both linear and quadratic inequalities is vital for a comprehensive understanding of algebraic relationships. A critical rule to remember is the reversal of the inequality symbol when multiplying or dividing by a negative number, which is essential for determining the correct solution set.