Steps for Solving Compound Inequalities

Inequalities are an essential part of mathematics, especially when dealing with real-life situations that involve ranges and boundaries. A compound inequality consists of two or more inequalities joined together by the words “and” or “or”. Solving compound inequalities requires a logical approach to find the values that satisfy the given conditions. In this article, we will discuss step-by-step instructions on how to solve compound inequalities.

Step 1: Understand the Given Problem
The first step in solving compound inequalities is to understand the problem statement. Read the given problem carefully and identify the compound inequality. Make sure you understand the symbols used in inequalities, such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

Step 2: Split the Compound Inequality
Breaking down the compound inequality into individual inequalities is crucial. If the compound inequality joins two inequalities by the word “and”, we split it into two separate inequalities. For instance, if we have the compound inequality 3x + 2 > 8 and x + 4 < 10, we would split it into 3x + 2 > 8 and x + 4 < 10. On the other hand, if the compound inequality joins two inequalities by the word "or", we split it into two separate inequalities and solve them separately. For example, if we have the compound inequality 2x + 3 ≥ 7 or x - 5 ≤ 2, we would split it into 2x + 3 ≥ 7 and x - 5 ≤ 2. Step 3: Solve the Individual Inequalities Now that we have split the compound inequality, we can solve the individual inequalities. Begin by solving each inequality as if it were a single inequality. Use the appropriate methods, such as adding, subtracting, multiplying, or dividing, to isolate the variable. Remember to apply the same operation to both sides of the inequality to maintain the balance. Step 4: Graph the Solutions on a Number Line After solving the individual inequalities, we need to graph the solutions on a number line. Draw a number line and mark the values that satisfy each inequality. Use an open circle (○) for inequalities that do not include the endpoint and a closed circle (●) for inequalities that include the endpoint. Draw an arrow to indicate the direction of the solution range. Step 5: Determine the Overlapping Solutions If the original compound inequality was split using the word "and", we need to determine the overlapping solutions. Identify the range that satisfies both individual inequalities by analyzing the overlapping portion on the number line. The overlapping range represents the solution to the compound inequality. Step 6: Combine the Solutions (if "or" was used) If the compound inequality was split using the word "or", we need to combine the solutions of the individual inequalities. This is done by using the word "or" to connect the solutions. For example, if the solutions to 2x + 3 ≥ 7 and x - 5 ≤ 2 are x ≥ 2 and x ≤ 7, we would combine them by stating x ≥ 2 or x ≤ 7. In conclusion, solving compound inequalities involves splitting the inequality, solving each individual inequality, graphing the solutions, and determining the overlapping or combined solutions. By following these step-by-step instructions, you can effectively solve compound inequalities and find the range of values that satisfy the given conditions.