Solving Compound Inequalities -18x + 21 > -15 OR 20x - 13 ≥ 27

In mathematics, inequalities play a crucial role in defining ranges and constraints for variables. When we encounter compound inequalities, which combine two or more inequalities with logical connectors like "OR" and "AND," the process of finding solutions becomes even more interesting. This article will delve into solving the compound inequality $-18x + 21 > -15$ OR $20x - 13 \geq 27$, providing a step-by-step approach, detailed explanations, and practical insights to help you master this fundamental concept.

Understanding Compound Inequalities

Compound inequalities are mathematical statements that combine two or more inequalities using logical connectives such as "OR" or "AND." These connectives dictate how the solutions of the individual inequalities combine to form the solution set of the compound inequality. The "OR" connective means that the solution set includes all values that satisfy either one or both of the inequalities. Conversely, the "AND" connective means that the solution set includes only the values that satisfy both inequalities simultaneously. Understanding these connectives is crucial for accurately solving and interpreting compound inequalities.

Step-by-Step Solution

Solving the First Inequality: $-18x + 21 > -15$

To solve the first inequality, $-18x + 21 > -15$, we need to isolate the variable $x$. This involves performing algebraic operations on both sides of the inequality while maintaining its balance. Here’s a detailed breakdown of the steps:

1. Isolate the term with x: First, we subtract 21 from both sides of the inequality to isolate the term containing $x$:

   $$-18x + 21 - 21 > -15 - 21$$

   $$-18x > -36$$

2. Divide by the coefficient of x: Next, we divide both sides by -18 to solve for $x$. It's crucial to remember that when dividing or multiplying an inequality by a negative number, we must reverse the inequality sign:

   $$\frac{-18x}{-18} < \frac{-36}{-18}$$

   $$x < 2$$

Therefore, the solution to the first inequality is $x < 2$. This means that any value of $x$ less than 2 satisfies the first part of the compound inequality.

Solving the Second Inequality: $20x - 13 \geq 27$

Now, let's solve the second inequality, $20x - 13 \geq 27$. Similar to the first inequality, we'll isolate $x$ by performing algebraic operations on both sides:

1. Isolate the term with x: Add 13 to both sides of the inequality:

   $$20x - 13 + 13 \geq 27 + 13$$

   $$20x \geq 40$$

2. Divide by the coefficient of x: Divide both sides by 20:

   $$\frac{20x}{20} \geq \frac{40}{20}$$

   $$x \geq 2$$

Thus, the solution to the second inequality is $x \geq 2$. This means that any value of $x$ greater than or equal to 2 satisfies the second part of the compound inequality.

Combining the Solutions with "OR"

Since the original problem uses the connective "OR," we need to find the union of the solutions to the two inequalities. The first inequality gives us $x < 2$, and the second gives us $x \geq 2$. The union of these two solution sets includes all values that satisfy either $x < 2$ or $x \geq 2$.

When we combine these two conditions, we observe that every real number is included. The first inequality covers all numbers less than 2, while the second inequality covers 2 and all numbers greater than 2. Together, they cover the entire number line.

Therefore, the solution to the compound inequality $-18x + 21 > -15$ OR $20x - 13 \geq 27$ is all real numbers.

Representing the Solution

Interval Notation

The solution can be represented in interval notation as $(-\infty, \infty)$, indicating that all real numbers are included in the solution set. This notation is a concise way to express an unbounded set of numbers.

Number Line Representation

Graphically, we can represent the solution on a number line. For $x < 2$, we draw an open circle at 2 and shade the line to the left, indicating all values less than 2 are included. For $x \geq 2$, we draw a closed circle at 2 and shade the line to the right, indicating that 2 and all values greater than 2 are included. Since the solution is the union of these two sets, the entire number line is shaded, visually confirming that all real numbers are solutions.

Common Mistakes and How to Avoid Them

Forgetting to Reverse the Inequality Sign

One common mistake is forgetting to reverse the inequality sign when multiplying or dividing by a negative number. This can lead to an incorrect solution set. Always remember to flip the inequality sign when performing these operations.

For example, in the inequality $-18x > -36$, dividing both sides by -18 requires flipping the “>” sign to “<”, resulting in $x < 2$.

Misinterpreting Compound Inequalities

Another common error is misinterpreting the connectives "OR" and "AND." "OR" means that the solution must satisfy at least one of the inequalities, while "AND" means the solution must satisfy both inequalities simultaneously. Understanding this distinction is crucial for accurately determining the solution set.

Arithmetic Errors

Simple arithmetic errors can also lead to incorrect solutions. Double-check your calculations, especially when dealing with negative numbers and fractions.

Practice Problems

To reinforce your understanding, try solving these practice problems:

1. Solve for $x$: $5x - 3 < 7$ OR $2x + 1 > 9$
2. Solve for $y$: $-3y + 4 \geq 10$ OR $4y - 5 \leq 3$
3. Solve for $z$: $6z - 2 < 16$ OR $-2z + 5 > 1$

Conclusion

Solving compound inequalities involves breaking down the problem into individual inequalities, solving each one separately, and then combining the solutions based on the logical connective (OR or AND). For the compound inequality $-18x + 21 > -15$ OR $20x - 13 \geq 27$, we found that the solution is all real numbers, represented as $(-\infty, \infty)$ in interval notation. By understanding the steps involved and avoiding common mistakes, you can confidently tackle compound inequalities in various mathematical contexts.

This comprehensive guide has equipped you with the knowledge and skills necessary to solve compound inequalities effectively. Keep practicing, and you’ll master this important mathematical concept.