Mastering Linear Inequalities: A Simple Guide

Mastering Linear Inequalities: A Simple Guide

Linear inequalities can sometimes feel like a puzzle, especially when they involve variables on both sides. But don't worry! With a systematic approach, you can conquer them. Let's take the inequality $9t + 7 > -9t - 6$ as our example and break down how to solve it step-by-step. This process will not only help you find the solution for this specific problem but will also equip you with the skills to tackle a wide range of similar linear inequalities.

Understanding the Goal: Isolating the Variable

When we solve any equation or inequality, our primary goal is to isolate the variable (in this case, 't') on one side of the inequality sign. Think of it like trying to get all the 't' terms together and all the constant numbers together. This makes it easier to see what 't' is greater than, less than, or equal to. In our example, $9t + 7 > -9t - 6$, we have 't' terms on both the left side ($9t$) and the right side ($-9t$), and constants on both sides (7 and -6). Our mission is to move all the 't' terms to one side and all the constants to the other.

Step 1: Gathering the Variable Terms

The first strategic move is to combine all the terms containing the variable 't'. We can do this by performing the opposite operation on both sides of the inequality to eliminate a term from one side. For $9t + 7 > -9t - 6$, we see a $-9t$ on the right side. To eliminate it from the right and move it to the left, we'll add $9t$ to both sides. Remember, whatever you do to one side of an inequality, you must do to the other to maintain the balance. So, we add $9t$ to both $9t + 7$ and $-9t - 6$:

$(9t + 7) + 9t > (-9t - 6) + 9t$

This simplifies to:

$18t + 7 > -6$

Now, all our 't' terms are on the left side. This is a significant step forward in isolating 't'. If we had chosen to subtract $9t$ from both sides, we would have ended up with $7 > -18t - 6$, which is also a valid starting point, but dealing with a positive coefficient for 't' (like $18t$) is often less prone to errors when dividing later.

Step 2: Consolidating the Constant Terms

With the variable terms consolidated, our next objective is to gather all the constant numbers on the opposite side of the inequality. In our current inequality, $18t + 7 > -6$, we have the constant term $+7$ on the left side with the $18t$. To move this $+7$ to the right side, we perform the opposite operation, which is subtracting 7 from both sides of the inequality:

$18t + 7 - 7 > -6 - 7$

This simplifies nicely to:

$18t > -13$

At this stage, we have successfully separated the variable term ($18t$) from the constant term ($-13$). The inequality now clearly shows the relationship between $18t$ and $-13$. We are one step away from finding the value of 't' itself.

Step 3: Isolating the Variable Completely

The final step to solve for 't' is to eliminate the coefficient multiplying it. Currently, 't' is multiplied by 18. To isolate 't', we need to divide both sides of the inequality by 18. A crucial rule to remember with inequalities is that if you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign. In this case, we are dividing by a positive number (18), so the inequality sign ($>$) remains the same.

rac{18t}{18} > rac{-13}{18}

This gives us our final solution:

t > -rac{13}{18}

This means that any value of 't' that is greater than -rac{13}{18} will satisfy the original inequality $9t + 7 > -9t - 6$. For example, if $t=0$, $9(0)+7 > -9(0)-6$, which is $7 > -6$, true. If $t=-1$, $9(-1)+7 > -9(-1)-6$, which is $-2 > 3$, false. This confirms our solution.

Verification and Understanding the Solution

Solving inequalities involves more than just algebraic manipulation; it's also about understanding what the solution represents. The inequality t > -rac{13}{18} means that the solution is not a single value but an infinite set of values. On a number line, this would be represented by an open circle at -rac{13}{18} and shading to the right, indicating all numbers greater than -rac{13}{18}.

To verify our solution, we can pick a value for 't' that is greater than -rac{13}{18} (like $t=0$) and substitute it back into the original inequality. We already did this, and it checked out: $9(0) + 7 > -9(0) - 6 ightarrow 7 > -6$, which is true.

We can also pick a value for 't' that is not greater than -rac{13}{18} (like $t=-1$, which is less than -rac{13}{18}) and substitute it back. This should result in a false statement. We saw that $9(-1) + 7 > -9(-1) - 6 ightarrow -2 > 3$, which is false. This confirms that our boundary is correct and that values less than -rac{13}{18} do not satisfy the inequality.

Understanding how to solve linear inequalities like $9t + 7 > -9t - 6$ is a fundamental skill in algebra. It builds a foundation for more complex mathematical concepts, including graphing linear equations and understanding functions. By systematically isolating the variable and remembering the rules for manipulating inequalities, you can confidently solve any such problem. For more practice and deeper understanding, resources like Khan Academy offer excellent tutorials and exercises. Another great resource for mathematical understanding is Brilliant.org. Keep practicing, and you'll find these problems become second nature!

Conclusion

Solving the linear inequality $9t + 7 > -9t - 6$ involves straightforward algebraic steps: gather variable terms, consolidate constant terms, and finally, isolate the variable. The solution, t > -rac{13}{18}, represents an infinite set of numbers. Mastering this process equips you to tackle similar problems and strengthens your algebraic foundation.