Solving Linear Inequalities A Comprehensive Guide To -5x + 7 > 42

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In the realm of mathematics, solving inequalities is a fundamental skill, particularly crucial in algebra and calculus. Inequalities, unlike equations, deal with relationships where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. This article delves into the intricacies of solving the linear inequality -5x + 7 > 42, providing a step-by-step guide and highlighting key concepts to ensure a thorough understanding.

Understanding Inequalities

Before we dive into the solution, let’s clarify what inequalities are and how they differ from equations. An inequality is a mathematical statement that compares two expressions using inequality symbols such as >, <, ≥, or ≤. For instance, 'a > b' means 'a' is greater than 'b,' while 'a < b' means 'a' is less than 'b.' The symbols '≥' and '≤' represent 'greater than or equal to' and 'less than or equal to,' respectively.

Linear inequalities, specifically, are inequalities that involve a linear expression. A linear expression is an algebraic expression in which the highest power of the variable is 1. The inequality -5x + 7 > 42 is a linear inequality because the variable 'x' is raised to the power of 1.

Key Properties of Inequalities

When solving inequalities, it's essential to understand the properties that govern them. These properties dictate how we can manipulate inequalities while preserving their validity. The main properties include:

  1. Addition Property: Adding the same number to both sides of an inequality does not change the inequality's direction. If a > b, then a + c > b + c.
  2. Subtraction Property: Subtracting the same number from both sides of an inequality does not change the inequality's direction. If a > b, then a - c > b - c.
  3. Multiplication Property (Positive Number): Multiplying both sides of an inequality by the same positive number does not change the inequality's direction. If a > b and c > 0, then ac > bc.
  4. Division Property (Positive Number): Dividing both sides of an inequality by the same positive number does not change the inequality's direction. If a > b and c > 0, then a/c > b/c.
  5. Multiplication Property (Negative Number): Multiplying both sides of an inequality by the same negative number reverses the inequality's direction. If a > b and c < 0, then ac < bc.
  6. Division Property (Negative Number): Dividing both sides of an inequality by the same negative number reverses the inequality's direction. If a > b and c < 0, then a/c < b/c.

The crucial point to remember is that multiplying or dividing by a negative number flips the direction of the inequality. This is a common area for errors, so it’s worth emphasizing.

Step-by-Step Solution of -5x + 7 > 42

Now, let's apply these properties to solve the inequality -5x + 7 > 42 step by step. Our goal is to isolate the variable 'x' on one side of the inequality.

Step 1: Isolate the Term with 'x'

The first step involves isolating the term that contains 'x,' which in this case is -5x. To do this, we need to eliminate the '+7' on the left side of the inequality. We can achieve this by subtracting 7 from both sides. This utilizes the subtraction property of inequalities.

-5x + 7 > 42

Subtract 7 from both sides:

-5x + 7 - 7 > 42 - 7

This simplifies to:

-5x > 35

Step 2: Isolate 'x'

Now that we have -5x > 35, our next goal is to isolate 'x.' To do this, we need to get rid of the '-5' coefficient. Since -5 is multiplying 'x,' we will divide both sides of the inequality by -5. Remember, this is where the crucial rule about negative numbers comes into play. Dividing by a negative number reverses the direction of the inequality.

-5x > 35

Divide both sides by -5:

(-5x) / -5 < 35 / -5

Notice that the '>' symbol has changed to '<' because we divided by a negative number. This simplifies to:

x < -7

Step 3: Interpret the Solution

Our solution is x < -7. This means that 'x' can be any number less than -7. In other words, the solution set includes all real numbers that are strictly less than -7. To visualize this, we can represent the solution on a number line.

Representing the Solution on a Number Line

On a number line, we would draw an open circle at -7 to indicate that -7 is not included in the solution set (since x is strictly less than -7). Then, we would shade the line to the left of -7, representing all the numbers that are less than -7.

Expressing the Solution in Interval Notation

Another way to express the solution is using interval notation. Interval notation uses parentheses and brackets to indicate the range of values that satisfy the inequality. For x < -7, the interval notation is (-∞, -7). The parenthesis indicates that -7 is not included in the interval, and -∞ represents negative infinity, indicating that the solution extends indefinitely in the negative direction.

Common Mistakes to Avoid

Solving inequalities involves several steps, and it's easy to make mistakes along the way. Here are some common pitfalls to watch out for:

  1. Forgetting to Reverse the Inequality: As emphasized earlier, one of the most common mistakes is forgetting to reverse the inequality sign when multiplying or dividing by a negative number. Always double-check this step when dealing with negative coefficients.

  2. Incorrectly Applying the Order of Operations: Ensure you follow the correct order of operations (PEMDAS/BODMAS) when simplifying both sides of the inequality.

  3. Making Arithmetic Errors: Simple arithmetic errors can lead to incorrect solutions. Take your time and double-check your calculations.

  4. Misinterpreting the Solution Set: Understand what the solution means in the context of the problem. For example, x < -7 means all numbers less than -7, not just -7 itself.

  5. Incorrectly Graphing the Solution: When representing the solution on a number line, be mindful of whether to use an open or closed circle and which direction to shade.

Practice Problems

To solidify your understanding, let's work through a few practice problems:

Problem 1: Solve 3x - 5 ≤ 10

Solution:

  1. Add 5 to both sides: 3x ≤ 15
  2. Divide both sides by 3: x ≤ 5

The solution set is all numbers less than or equal to 5. In interval notation, this is (-∞, 5].

Problem 2: Solve -2x + 4 < 8

Solution:

  1. Subtract 4 from both sides: -2x < 4
  2. Divide both sides by -2 (and reverse the inequality): x > -2

The solution set is all numbers greater than -2. In interval notation, this is (-2, ∞).

Problem 3: Solve 4x + 2 > 6x - 8

Solution:

  1. Subtract 4x from both sides: 2 > 2x - 8
  2. Add 8 to both sides: 10 > 2x
  3. Divide both sides by 2: 5 > x

This can also be written as x < 5. The solution set is all numbers less than 5. In interval notation, this is (-∞, 5).

Applications of Inequalities

Inequalities are not just abstract mathematical concepts; they have practical applications in various fields. Here are a few examples:

  1. Budgeting: Inequalities can be used to represent budget constraints. For example, if you have a budget of $100 and want to buy items costing 'x' and 'y' dollars, the inequality x + y ≤ 100 can represent your spending limit.

  2. Physics: Inequalities are used to describe ranges of physical quantities. For instance, the speed of an object might be constrained by an inequality such as v ≤ 100 m/s, representing a maximum speed.

  3. Optimization: In optimization problems, inequalities often define constraints on the variables. For example, in linear programming, inequalities are used to define the feasible region.

  4. Statistics: Inequalities are used in statistical hypothesis testing to define critical regions.

Advanced Topics in Inequalities

While this article focuses on linear inequalities, there are more advanced topics in inequalities that are worth mentioning:

  1. Quadratic Inequalities: These involve quadratic expressions and require techniques such as factoring or using the quadratic formula to find critical points.

  2. Rational Inequalities: These involve rational expressions and require careful consideration of the signs of the numerator and denominator.

  3. Absolute Value Inequalities: These involve absolute value expressions and require breaking the problem into cases.

  4. Systems of Inequalities: These involve multiple inequalities and require finding the region that satisfies all inequalities simultaneously.

Conclusion

Solving inequalities is a crucial skill in mathematics with wide-ranging applications. By understanding the properties of inequalities and following a step-by-step approach, you can confidently solve linear inequalities and tackle more complex problems. Remember to pay special attention to reversing the inequality sign when multiplying or dividing by a negative number, and always double-check your work to avoid common mistakes. With practice, you'll become proficient in solving inequalities and appreciate their significance in various mathematical and real-world contexts. Solving the inequality -5x + 7 > 42, we have demonstrated a clear, methodical process, arriving at the solution x < -7. This comprehensive guide should equip you with the knowledge and skills to solve similar problems and appreciate the broader applications of inequalities in mathematics and beyond.