How to Solve Inequalities

Understanding Inequalities

Inequalities are mathematical expressions involving symbols such as <, >, ≤, and ≥. They are used to compare two values or expressions. Solving inequalities involves finding the values of variables that satisfy the inequality condition. In this article, we will explore several methods to solve inequalities effectively.

Basic Methods for Solving Inequalities

Before diving into complex strategies, it's essential to grasp the basic principles of solving inequalities:

Just like equations, you can add or subtract the same value from both sides of an inequality without changing its direction. For example, if you have x + 3 > 5, subtracting 3 from both sides gives you x > 2.

When multiplying or dividing both sides of an inequality by a positive number, the direction of the inequality remains unchanged. However, if you multiply or divide by a negative number, you must flip the inequality sign. For example, if you have -2x > 6, dividing both sides by -2 yields x < -3.

Graphical Solutions

Graphing is a powerful tool to visualize and solve inequalities. By plotting the inequality on a number line or coordinate plane, you can easily identify the solution set.

For simple inequalities, use a number line to graph the solution. For example, to represent x > 2, draw an open circle at 2 and shade all numbers greater than 2.

For inequalities involving two variables, such as y < 2x + 3, graph the corresponding equation y = 2x + 3 as a boundary line. Then, test a point not on the line to determine which side of the line to shade.

Compound Inequalities

Compound inequalities involve two or more simple inequalities joined by "and" or "or". Solving them requires handling each part of the inequality separately.

For inequalities like 1 < x < 5, find values of x that satisfy both conditions simultaneously. Combine the solutions to get the intersection of the solution sets.

For inequalities such as x < -2 or x > 3, find values that satisfy at least one condition. The solution is the union of the individual solution sets.

Quadratic Inequalities

Quadratic inequalities involve expressions with terms like x². To solve them, follow these steps:

Set the quadratic expression equal to zero and solve for the variable. For example, to solve x² - 3x - 10 > 0, first solve x² - 3x - 10 = 0 to find critical points.

Use the critical points to divide the number line into intervals. Test a point in each interval to determine whether it satisfies the inequality.

Rational Inequalities

Rational inequalities involve ratios of polynomial expressions. Solving them requires finding critical points and testing intervals, similar to quadratic inequalities.

Set the numerator and denominator equal to zero separately to find critical points. These points help identify interval boundaries.

Determine where the expression is positive or negative by picking test points in each interval. Consider the sign changes at critical points.

Absolute Value Inequalities

Absolute value inequalities involve expressions with absolute value bars. They require splitting into separate cases based on the definition of absolute value.

For an inequality like |x - 3| < 5, solve x - 3 < 5 and x - 3 > -5 separately. Combine the solutions to find the solution set.

Each case results in a simple inequality that can be solved using basic methods. The final solution is the intersection or union of these solutions, depending on the initial inequality.