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INHOW TO SOLVE INEQUALITIES?
Solving inequalities involves finding the range of values that satisfy a given mathematical expression. Here’s a breakdown of the process and some examples:
Basic Principles:
Similar to Equations: Many of the same operations used to solve equations apply to inequalities. You can add, subtract, multiply, or divide both sides of an inequality.
Flipping the Inequality: If you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.
Number Lines: Visualizing solutions on a number line can be very helpful and Solutions are often expressed using interval notation.
Steps to Solve Inequalities:
Simplify: Combine like terms and eliminate parentheses.
Isolate the Variable: Use inverse operations to isolate the variable on one side of the inequality.
Consider Negative Multiplication/Division: If you multiply or divide by a negative number, remember to reverse the inequality sign.
Express the Solution: Write the solution using inequality notation or interval notation.
Examples:
Example 1: 2x + 3 > 7
Subtract 3 from both sides: 2x > 4
Divide both sides by 2: x > 2
Solution: x > 2 (or in interval notation, (2, ∞))
Example 2: -3x + 5 ≤ 14
Subtract 5 from both sides: -3x ≤ 9
Divide both sides by -3 (and reverse the inequality): x ≥ -3
Solution: x ≥ -3 (or in interval notation, [-3, ∞))
Example 3: Compound Inequalities
Solve: -1 < 2x + 3 < 9
Subtract 3 from all parts: -4 < 2x < 6
Divide all parts by 2: -2 < x < 3
Solution: -2 < x < 3 (or in interval notation, (-2, 3))
Types of Inequalities:
Linear Inequalities: Involve variables raised to the first power.
Quadratic Inequalities: Involve variables raised to the second power.
Polynomial Inequalities: Involve variables raised to higher powers.
Rational Inequalities: Involve ratios of polynomials.
Absolute Value Inequalities: Involve absolute value expressions.
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