Master Solving Inequalities: A Step-by-Step Guide

Welcome! Today, we're diving into the world of inequalities, specifically tackling an expression like 9t + 7 > -9t - 6. Don't let the symbols and variables intimidate you; solving inequalities is a skill that, once mastered, opens up a whole new way of understanding mathematical relationships. Think of it like solving equations, but with a slight twist: instead of finding a single solution, you're often finding a range of values that satisfy the condition. This guide will break down the process, making it accessible and even enjoyable. We'll start with the basics of what inequalities are, move through the steps to solve them, and ensure you feel confident in your ability to find the solution set for expressions like our example. Let’s get started on this journey to understanding how to solve inequalities effectively.

Understanding the Core Concept of Inequalities

Before we jump into solving specific problems, let's solidify our understanding of what inequalities represent. Unlike equations, which state that two expressions are equal (e.g., a = b), inequalities express a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. The symbols we commonly see are: > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Our example, 9t + 7 > -9t - 6, uses the 'greater than' symbol. This means we are looking for all the values of t that make the expression on the left side (9t + 7) larger than the expression on the right side (-9t - 6).

When we solve an equation, we aim to find the specific value(s) of the variable that make the equation true. For inequalities, however, the solution is typically a range of values. For instance, if we solved an inequality and found that t > 5, it means that any number greater than 5 (like 5.1, 10, or 1000) will satisfy the original inequality. This concept of a solution set is crucial. We often represent these solution sets on a number line, using open circles for strict inequalities (> and <) and closed circles for inclusive inequalities (≥ and ≤), with shading to indicate the range of valid numbers. Understanding this fundamental difference between equations and inequalities is the first step towards mastering how to solve them. It’s about identifying conditions rather than exact points. The process of manipulating an inequality to isolate the variable shares many similarities with solving equations, but there's one key difference that we'll explore next: the behavior when multiplying or dividing by a negative number. This distinction is vital for ensuring your solution accurately reflects the inequality's relationship.

Step-by-Step: Solving 9t + 7 > -9t - 6

Now, let's roll up our sleeves and tackle the inequality 9t + 7 > -9t - 6. The primary goal here, just like with equations, is to isolate the variable t on one side of the inequality sign. We'll use the same algebraic operations—addition, subtraction, multiplication, and division—but we must be mindful of a special rule. First, let's gather all the terms containing t on one side. To do this, we can add 9t to both sides of the inequality. This is a valid operation because adding the same quantity to both sides does not change the direction of the inequality.

9t + 7 + 9t > -9t - 6 + 9t

This simplifies to:

18t + 7 > -6

Next, we want to isolate the t term by moving the constant term (+7) to the other side. We achieve this by subtracting 7 from both sides:

18t + 7 - 7 > -6 - 7

Which gives us:

18t > -13

We are now very close to finding the solution for t. The final step is to divide both sides by 18 to get t by itself. Since 18 is a positive number, the direction of the inequality sign remains unchanged.

18t / 18 > -13 / 18

And the solution is:

t > -13/18

This means that any value of t that is greater than -13/18 will make the original inequality 9t + 7 > -9t - 6 true. For example, if t = 0 (which is greater than -13/18), the left side is 9(0) + 7 = 7 and the right side is -9(0) - 6 = -6. Since 7 > -6, the inequality holds. If we picked a value less than -13/18, say t = -1, the left side becomes 9(-1) + 7 = -2 and the right side becomes -9(-1) - 6 = 9 - 6 = 3. Here, -2 is not greater than 3, so the inequality does not hold. This step-by-step approach, focusing on maintaining the inequality's balance, is key to correctly solving these problems.

The Crucial Rule: Multiplying or Dividing by Negatives

Let's delve into the most critical aspect of solving inequalities that often trips people up: what happens when you multiply or divide both sides by a negative number. Remember the example we solved, 9t + 7 > -9t - 6, which led to t > -13/18? This was straightforward because all the numbers we divided or multiplied by were positive. However, inequalities have a unique rule: if you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. Why? Let's consider a simple inequality: 2 < 5. This is clearly true. Now, what happens if we multiply both sides by -1?

If we don't reverse the sign, we'd get -2 < -5. But this is false! -2 is actually greater than -5. To make the statement true, we must reverse the inequality sign: -2 > -5. This demonstrates why the rule exists. Multiplying by a negative number effectively flips the position of numbers on the number line relative to each other, thus requiring the inequality sign to flip as well.

Let's illustrate this with a slightly modified example. Suppose we needed to solve -3x < 12. To isolate x, we would divide both sides by -3. Since we are dividing by a negative number, we must reverse the inequality sign from < to >:

-3x / -3 > 12 / -3

x > -4

If we had forgotten to reverse the sign, we would have incorrectly concluded that x < -4. Let's test this. If x = -5 (which is less than -4), the original inequality -3x < 12 becomes -3(-5) < 12, which is 15 < 12. This is false. However, if we use the correct solution x > -4, let's test x = -3 (which is greater than -4). The original inequality becomes -3(-3) < 12, which is 9 < 12. This is true. This rule is fundamental and applies universally whenever you perform a multiplication or division by a negative value on either side of the inequality. Always double-check this step to ensure your solution is accurate. Understanding this nuance is key to confidently solving any inequality problem you encounter.

Representing Solutions on a Number Line

Once you've solved an inequality and found the range of values for your variable (like t > -13/18), visualizing this solution set is incredibly helpful. The standard way to do this is by using a number line. A number line is simply a line with numbers marked at intervals, extending infinitely in both directions. To represent our solution t > -13/18, we first locate the value -13/18 on the number line. Since -13/18 is between -1 and 0 (closer to -1), we'd find that spot.

Next, we need to indicate that t must be greater than this value. This means we place an open circle at -13/18. An open circle signifies that the number itself is not included in the solution set. If our inequality had been t ≥ -13/18 (greater than or equal to), we would use a closed circle (or a filled-in circle) to indicate that -13/18 is included in the solution. Finally, since t must be greater than -13/18, we shade the portion of the number line to the right of the open circle. This shading represents all the numbers that satisfy the condition t > -13/18.

Let's consider another example. If we solved an inequality and got x ≤ 3. We would locate 3 on the number line. Since the inequality is 'less than or equal to', we would place a closed circle at 3. Then, because x must be less than or equal to 3, we would shade the portion of the number line to the left of the closed circle. This shaded region, including the point at 3, represents all possible values of x.

Number line representations are powerful tools. They provide a clear, visual confirmation of your algebraic solution and help in understanding the scope of possible answers. They are particularly useful when dealing with compound inequalities (which involve two inequality conditions combined) or when comparing solution sets from different problems. Mastering this visualization technique complements your algebraic skills, offering a comprehensive understanding of inequality solutions. For further exploration on number line representations, the resources at Khan Academy offer excellent visual aids and practice exercises.

Conclusion

We've journeyed through the essentials of solving inequalities, using 9t + 7 > -9t - 6 as our guiding example. We learned that inequalities express relationships of comparison rather than equality, and their solutions often represent a range of values. The key steps involved were isolating the variable t through algebraic manipulation, much like solving equations. Crucially, we emphasized the special rule: reversing the inequality sign whenever multiplying or dividing by a negative number. Finally, we explored how to visually represent these solution sets on a number line using open or closed circles and shading. By applying these principles, you can confidently tackle a wide variety of inequality problems. For additional practice and a deeper dive into related concepts, exploring resources like Math is Fun can be highly beneficial.