Solve 9t + 7 > -9t - 6: A Simple Guide

Inequalities can sometimes look a little intimidating, especially when they involve variables on both sides of the greater than (>) or less than (<) sign. But don't worry, solving them is much like solving regular equations. Today, we're going to break down how to solve the inequality 9t + 7 > -9t - 6 step-by-step. This will not only give you the solution but also build your confidence in tackling similar problems.

Understanding the Goal: Isolating the Variable

Our main objective when solving any inequality or equation is to isolate the variable – in this case, 't' – on one side of the inequality sign. This means we want to get 't' all by itself, so we can see what values of 't' make the original statement true. To do this, we'll use a set of rules that are very similar to those used for solving equations. The key difference with inequalities is that if we multiply or divide both sides by a negative number, we must flip the direction of the inequality sign. Thankfully, in this particular problem, we won't need to do that, which makes things a bit simpler.

To start isolating 't', we need to get all the terms containing 't' onto one side of the inequality and all the constant terms (numbers without a variable) onto the other side. It often makes things easier if you aim to have a positive coefficient for your variable in the end. Let's look at our inequality: 9t + 7 > -9t - 6. We have '9t' on the left and '-9t' on the right. To get them together, we can add '9t' to both sides. Why add? Because adding '9t' to '-9t' will result in 0, and adding '9t' to '9t' will give us '18t', which is a positive coefficient. This is a good first move.

So, let's add '9t' to both sides:

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

Combine the 't' terms on the left: 9t + 9t equals 18t.

Combine the 't' terms on the right: -9t + 9t equals 0t, or simply 0.

Our inequality now looks like this: 18t + 7 > -6.

We're one step closer! Now, we have the 't' term on the left and a constant term ('7') also on the left. We need to move that constant term to the right side. To do this, we perform the opposite operation of adding 7, which is subtracting 7. We must do this to both sides of the inequality to maintain the balance.

Subtract 7 from both sides:

18t + 7 - 7 > -6 - 7

On the left side, 7 - 7 is 0, leaving us with 18t.

On the right side, -6 - 7 equals -13.

So, the inequality simplifies to: 18t > -13.

We're in the home stretch! The variable 't' is now multiplied by 18. To isolate 't', we need to perform the inverse operation of multiplication, which is division. We will divide both sides of the inequality by 18. Since 18 is a positive number, we do not need to flip the inequality sign.

Divide both sides by 18:

18t / 18 > -13 / 18

This leaves us with t > -13/18.

And there you have it! The solution to the inequality 9t + 7 > -9t - 6 is t > -13/18. This means any value of 't' that is strictly greater than -13/18 will make the original inequality true. This process of isolating the variable is fundamental to solving all linear inequalities and equations.

Step-by-Step Solution Breakdown

Let's recap the process for solving 9t + 7 > -9t - 6 to ensure every step is crystal clear. This methodical approach is key to avoiding errors and building confidence when you encounter more complex inequalities.

Step 1: Combine Variable Terms

Our first goal is to gather all terms containing the variable 't' on one side of the inequality. We have 9t on the left and -9t on the right. To move the -9t from the right to the left, we perform the opposite operation: addition. We add 9t to both sides of the inequality.

Original Inequality: 9t + 7 > -9t - 6

Add 9t to both sides: 9t + 7 + 9t > -9t - 6 + 9t

Simplify: 18t + 7 > -6

By doing this, we've successfully moved all 't' terms to the left side and simplified the expression, setting the stage for the next step. It's important to remember that whatever operation you perform on one side of the inequality, you must perform the same operation on the other side to maintain the truth of the statement.

Step 2: Combine Constant Terms

Now that our variable terms are consolidated, we need to isolate them by moving all the constant terms (numbers without variables) to the other side. Currently, we have +7 on the left side with our 18t. To move it, we perform the opposite operation: subtraction. We subtract 7 from both sides of the inequality.

Inequality after Step 1: 18t + 7 > -6

Subtract 7 from both sides: 18t + 7 - 7 > -6 - 7

Simplify: 18t > -13

This step effectively separates the variable term from the numerical constants, bringing us even closer to our final solution. The core principle here is maintaining equality (or in this case, inequality) by applying inverse operations symmetrically across the inequality sign.

Step 3: Isolate the Variable

The final step in solving for 't' is to undo the multiplication. Our variable 't' is currently multiplied by 18. To isolate 't', we perform the inverse operation of multiplication, which is division. We divide both sides of the inequality by 18.

Inequality after Step 2: 18t > -13

Divide both sides by 18: 18t / 18 > -13 / 18

Simplify: t > -13/18

Since we divided by a positive number (18), the direction of the inequality sign does not change. If we had divided by a negative number, we would have had to flip the > to a <.

The Solution:

t > -13/18

This result tells us that any number greater than -13/18 will satisfy the original inequality. For example, if t = 0 (which is greater than -13/18), let's check: 9(0) + 7 > -9(0) - 6 0 + 7 > 0 - 6 7 > -6 This is true! Let's try a number smaller than -13/18, say t = -1 (which is -18/18, smaller than -13/18). 9(-1) + 7 > -9(-1) - 6 -9 + 7 > 9 - 6 -2 > 3 This is false, as expected. This verification process helps confirm that our solution is correct.

Visualizing the Solution on a Number Line

Understanding the solution t > -13/18 becomes much clearer when we visualize it on a number line. A number line is a graphical representation of numbers, with integers typically spaced equally. It helps us grasp the concept of