Solve The Inequality:$\[ |6t - 7| - 8 \geq 3 \\]

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Introduction

In this article, we will delve into the world of inequalities and learn how to solve a specific type of inequality involving absolute values. The given inequality is $|6t - 7| - 8 \geq 3$, where we need to isolate the variable $t$ and find the solution set that satisfies the inequality. We will break down the solution process into manageable steps, making it easier to understand and follow along.

Understanding Absolute Value Inequalities

Before we dive into solving the given inequality, let's take a moment to understand what absolute value inequalities are. An absolute value inequality is an inequality that involves the absolute value of an expression. The absolute value of a number is its distance from zero on the number line, without considering direction. In other words, the absolute value of a number is always non-negative.

For example, if we have the inequality $|x| \geq 2$, it means that the distance of $x$ from zero is greater than or equal to 2. This can be represented graphically as two separate inequalities: $x \geq 2$ and $x \leq -2$.

Solving the Inequality

Now that we have a basic understanding of absolute value inequalities, let's move on to solving the given inequality $|6t - 7| - 8 \geq 3$. To solve this inequality, we will follow the standard steps for solving absolute value inequalities.

Step 1: Add 8 to Both Sides

The first step in solving the inequality is to add 8 to both sides of the equation. This will help us isolate the absolute value expression.

${ |6t - 7| - 8 + 8 \geq 3 + 8 }$

Simplifying the left-hand side, we get:

${ |6t - 7| \geq 11 }$

Step 2: Split the Inequality into Two Separate Inequalities

Now that we have isolated the absolute value expression, we can split the inequality into two separate inequalities. This is because the absolute value of an expression is always non-negative, so we can split the inequality into two separate inequalities: one where the expression inside the absolute value is non-negative, and another where the expression inside the absolute value is negative.

${ 6t - 7 \geq 11 \quad \text{or} \quad 6t - 7 \leq -11 }$

Step 3: Solve the First Inequality

Let's start by solving the first inequality: $6t - 7 \geq 11$. To solve this inequality, we will add 7 to both sides of the equation.

${ 6t - 7 + 7 \geq 11 + 7 }$

Simplifying the left-hand side, we get:

${ 6t \geq 18 }$

Next, we will divide both sides of the equation by 6 to isolate the variable $t$.

${ \frac{6t}{6} \geq \frac{18}{6} }$

Simplifying the left-hand side, we get:

${ t \geq 3 }$

Step 4: Solve the Second Inequality

Now that we have solved the first inequality, let's move on to solving the second inequality: $6t - 7 \leq -11$. To solve this inequality, we will add 7 to both sides of the equation.

${ 6t - 7 + 7 \leq -11 + 7 }$

Simplifying the left-hand side, we get:

${ 6t \leq -4 }$

Next, we will divide both sides of the equation by 6 to isolate the variable $t$.

${ \frac{6t}{6} \leq \frac{-4}{6} }$

Simplifying the left-hand side, we get:

${ t \leq -\frac{2}{3} }$

Conclusion

In this article, we learned how to solve the inequality $|6t - 7| - 8 \geq 3$. We broke down the solution process into manageable steps, making it easier to understand and follow along. We started by adding 8 to both sides of the equation to isolate the absolute value expression. Then, we split the inequality into two separate inequalities and solved each one separately. Finally, we combined the solutions to the two inequalities to find the final solution set.

The final solution set is $t \geq 3$ or $t \leq -\frac{2}{3}$. This means that the variable $t$ can take on any value that is greater than or equal to 3, or any value that is less than or equal to $-\frac{2}{3}$.

Final Answer

The final answer is $\boxed{t \geq 3 \text{ or } t \leq -\frac{2}{3}}$.

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Introduction

In our previous article, we learned how to solve the inequality $|6t - 7| - 8 \geq 3$. We broke down the solution process into manageable steps, making it easier to understand and follow along. In this article, we will answer some of the most frequently asked questions about solving absolute value inequalities, including the given inequality.

Q&A

Q: What is the first step in solving an absolute value inequality?

A: The first step in solving an absolute value inequality is to isolate the absolute value expression. This can be done by adding or subtracting the same value from both sides of the equation.

Q: How do I split the inequality into two separate inequalities?

A: To split the inequality into two separate inequalities, you need to consider the two cases: when the expression inside the absolute value is non-negative, and when the expression inside the absolute value is negative. This can be done by setting up two separate inequalities: one where the expression inside the absolute value is greater than or equal to a certain value, and another where the expression inside the absolute value is less than or equal to a certain value.

Q: How do I solve the first inequality?

A: To solve the first inequality, you need to isolate the variable by adding or subtracting the same value from both sides of the equation. Then, you can divide both sides of the equation by the coefficient of the variable to get the final solution.

Q: How do I solve the second inequality?

A: To solve the second inequality, you need to isolate the variable by adding or subtracting the same value from both sides of the equation. Then, you can divide both sides of the equation by the coefficient of the variable to get the final solution.

Q: What is the final solution set for the inequality $|6t - 7| - 8 \geq 3$?

A: The final solution set for the inequality $|6t - 7| - 8 \geq 3$ is $t \geq 3$ or $t \leq -\frac{2}{3}$.

Q: Can I use a calculator to solve absolute value inequalities?

A: Yes, you can use a calculator to solve absolute value inequalities. However, it's always a good idea to check your work by hand to make sure you get the correct solution.

Q: What are some common mistakes to avoid when solving absolute value inequalities?

A: Some common mistakes to avoid when solving absolute value inequalities include:

• Not isolating the absolute value expression
• Not splitting the inequality into two separate inequalities
• Not solving each inequality separately
• Not combining the solutions to the two inequalities to get the final solution set

Conclusion

In this article, we answered some of the most frequently asked questions about solving absolute value inequalities, including the given inequality $|6t - 7| - 8 \geq 3$. We covered topics such as isolating the absolute value expression, splitting the inequality into two separate inequalities, and solving each inequality separately. We also discussed some common mistakes to avoid when solving absolute value inequalities.

Final Answer

The final answer is $\boxed{t \geq 3 \text{ or } t \leq -\frac{2}{3}}$.