Algebra Class: Solving Inequalities

Solving Inequalities

• Inequalities represent a relationship between two expressions that are not necessarily equal. Symbols used include: > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to), ≠ (not equal to).
• Solving inequalities follows similar steps to solving equations, but with a crucial difference: when multiplying or dividing both sides by a negative number, the inequality sign flips.
• For example, if -2x > 4, dividing both sides by -2 gives x < -2.

Graphing Inequalities on a Number Line

• Inequalities can be visualized on a number line.
• A solution set of an inequality is the set of all numbers that make the inequality true.
• An open circle (o) represents a value not included in the solution, while a closed circle (•) represents a value included in the solution.
• Graphing inequalities on a number line shows the range of values where the inequality holds true.

Compound Inequalities

• Compound inequalities combine two or more inequalities.
• They are joined by the words "and" or "or".
• "And" inequalities are satisfied only when all the conditions within the compound inequality are true. The solution represents the intersection of the solution sets of the individual inequalities.
• "Or" inequalities are satisfied when at least one of the conditions is true. The solution represents the union of the solution sets of the individual inequalities.

Intersection of Inequalities

• The intersection of two inequalities involves finding all values that satisfy both inequalities simultaneously.
• This corresponds to the region where the solution sets of the individual inequalities overlap.
• Graphically, it's the portion of the number line that is shaded in both graphs of the individual inequalities.
• Example: To find the intersection of x > 2 and x < 5, you are looking for the region where x is larger than 2 and smaller than 5, resulting in the solution set 2 < x < 5.

Union of Inequalities

• The union of two inequalities involves finding all values that satisfy at least one of the inequalities.
• This corresponds to the entire region shaded in the graphs of either inequality.
• Graphically, it's everything that is shaded in either graph.
• Example: To find the union of x < -1 and x > 3, it consists of any value less than -1 or greater than 3.

Applications of Inequalities

• Inequalities are essential in many real-world scenarios.
• For example, they can be used to express budget constraints, performance targets, or problem limits.
• They provide a way to quantify and express possible boundaries for variables in various contexts.
• Real-world problems involving constraints or conditions often translate naturally into inequalities.

Systems of Linear Inequalities

• A system of linear inequalities consists of two or more linear inequalities that have to be solved simultaneously.
• The solution to a system of linear inequalities is the set of all points that satisfy all of the given inequalities.
• Graphically, the solution is the region where all the shaded regions (corresponding to the solutions of the individual inequalities) overlap.
• This region will often form a polygonal shape or a unbounded region (in which case there would be an infinite number of solution points).

Solving Systems Graphically

• Graph each individual inequality in the system.
• Determine the overlapping solution region on the graph. This is the solution to the entire system.
• The coordinates of any point within the overlapping area are part of the solution set.

Solving Systems Algebraically (where possible)

• This approach is typically suitable for systems with two variables.
• The algebraic method involves solving one equation for a variable in terms of the other variable, then substituting this expression into the other equation.

Absolute Value Inequalities

• Absolute value inequalities are frequently solved by breaking them into two separate inequalities.
• For example, |x| < a is equivalent to -a < x < a, and |x| > a is equivalent to either x > a or x < -a.
• These methods are used strategically to simplify solving absolute value inequalities.

Interval Notation

• A shorthand way of representing an inequality is using interval notation.
• Specific symbols are used to denote open intervals ( ) or closed intervals [ ].
• For example, the interval (2,5) represents the inequality 2 < x < 5 and the interval [2,5] represents the inequality 2 ≤ x ≤ 5.
• This notation is commonly used in mathematics to concisely display solution sets.