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An inequality is a mathematical statement that compares two quantities or expressions using one of the inequality symbols (<, >, ≤, ≥).
The symbol < denotes "less than," > denotes "greater than," ≤ denotes "less than or equal to," and ≥ denotes "greater than or equal to."
Inequalities follow similar rules to equations, such as addition, subtraction, multiplication, and division.
When adding or subtracting a number on both sides of an inequality, the inequality symbol remains the same.
When multiplying or dividing both sides of an inequality by a positive number, the inequality symbol remains the same.
However, when multiplying or dividing both sides of an inequality by a negative number, the inequality symbol is reversed.
Simple rules for inequalities
Inequalities involving a modulus
Inequalities involving quotients
Inequalities involving square functions
To solve a simple inequality, the goal is to determine the range of values that satisfy the inequality statement.
Start by isolating the variable on one side of the inequality symbol. Similar to equations, you can perform addition, subtraction, multiplication, or division to achieve this.
Remember to reverse the inequality symbol if you multiply or divide both sides by a negative number.
Once the variable is isolated, express the solution as an inequality, specifying the range of values for which the inequality holds true.
A modulus refers to the absolute value of a number, often denoted by |x|.
Inequalities involving a modulus require considering both positive and negative values of the expression within the modulus signs.
When solving absolute value inequalities, create two separate inequalities, one with a positive modulus expression and the other with a negative modulus expression.
Solve each inequality separately and combine the solutions to determine the full range of values that satisfy the original inequality.
Inequalities involving quotients often require considering both positive and negative values of the variable.
To solve such inequalities, isolate the variable on one side of the expression.
When dividing both sides of the inequality by a positive number, the inequality symbol remains the same.
However, when dividing both sides by a negative number, the inequality symbol is reversed.
After obtaining the solution, express it as an inequality, indicating the range of values for which the inequality is true.
Inequalities involving square functions typically require factoring to find the critical points.
Start by setting the inequality to zero, giving you an equation involving a square function.
Factor the equation and find the critical points.
Test the regions determined by the critical points to determine the solution to the inequality.
Express the final solution as an inequality, specifying the valid range of values
Quadratic inequalities involve a quadratic expression or function and require finding the range of values that satisfy the inequality.
Similar to solving quadratic equations, you need to set the quadratic inequality to zero and factor it.
After factoring, you can determine the x-intercepts or critical points.
Test the intervals determined by the critical points or use a sign chart to determine the solution.
Express the solution as an inequality, indicating the valid range of values.