Select The Values That Make The Inequality $\frac{t}{8} \leq -7$ True. Then, Write An Equivalent Inequality In Terms Of $t$.(Numbers Are Written In Order From Least To Greatest Going Across.)Answer Attempt 1 Out Of

Understanding the Problem

In this problem, we are given an inequality $\frac{t}{8} \leq -7$ and asked to select the values that make it true. We will then write an equivalent inequality in terms of $t$. To solve this problem, we need to isolate the variable $t$ and find the range of values that satisfy the inequality.

Step 1: Multiply Both Sides by 8

To isolate $t$, we can start by multiplying both sides of the inequality by 8. This will eliminate the fraction and make it easier to work with.

$\frac{t}{8} \leq -7$

$8 \times \frac{t}{8} \leq 8 \times -7$

$t \leq -56$

Step 2: Write the Equivalent Inequality

Now that we have isolated $t$, we can write the equivalent inequality in terms of $t$. Since $t$ is less than or equal to $-56$, we can write the inequality as:

$t \leq -56$

Step 3: Select the Values that Make the Inequality True

To select the values that make the inequality true, we need to find the range of values that satisfy the inequality. Since $t$ is less than or equal to $-56$, the values that make the inequality true are all numbers less than or equal to $-56$.

The Final Answer

The values that make the inequality $\frac{t}{8} \leq -7$ true are all numbers less than or equal to $-56$. The equivalent inequality in terms of $t$ is:

$t \leq -56$

Example Solutions

To illustrate the concept, let's consider some example solutions.

• If $t = -50$, then $\frac{t}{8} = -6.25$, which is less than $-7$. Therefore, $t = -50$ is a solution to the inequality.
• If $t = -60$, then $\frac{t}{8} = -7.5$, which is less than $-7$. Therefore, $t = -60$ is a solution to the inequality.
• If $t = -70$, then $\frac{t}{8} = -8.75$, which is greater than $-7$. Therefore, $t = -70$ is not a solution to the inequality.

Conclusion

Q&A: Solving Inequalities

Q: What is an inequality?

A: An inequality is a statement that compares two expressions using a mathematical symbol, such as <, >, ≤, or ≥. Inequalities are used to describe relationships between variables and constants.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. You can do this by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

Q: What is the difference between an inequality and an equation?

A: An equation is a statement that says two expressions are equal, while an inequality is a statement that says one expression is greater than, less than, greater than or equal to, or less than or equal to another expression.

Q: How do I know which operation to perform on an inequality?

A: When solving an inequality, you can perform the same operations on both sides of the inequality that you would perform on an equation. However, if you multiply or divide both sides of the inequality by a negative number, you need to reverse the direction of the inequality sign.

Q: What is the order of operations for solving inequalities?

A: The order of operations for solving inequalities is the same as for solving equations:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I write an equivalent inequality?

A: To write an equivalent inequality, you need to perform the same operations on both sides of the inequality that you would perform on an equation. This will give you a new inequality that is equivalent to the original one.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that uses a strict inequality symbol, such as < or >. A non-strict inequality is an inequality that uses a non-strict inequality symbol, such as ≤ or ≥.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each inequality separately and then combine the solutions. A compound inequality is an inequality that contains two or more inequality symbols.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is $t \leq -56$. This means that the values that make the inequality $\frac{t}{8} \leq -7$ true are all numbers less than or equal to $-56$.

Example Solutions

To illustrate the concept, let's consider some example solutions.

• If $t = -50$, then $\frac{t}{8} = -6.25$, which is less than $-7$. Therefore, $t = -50$ is a solution to the inequality.
• If $t = -60$, then $\frac{t}{8} = -7.5$, which is less than $-7$. Therefore, $t = -60$ is a solution to the inequality.
• If $t = -70$, then $\frac{t}{8} = -8.75$, which is greater than $-7$. Therefore, $t = -70$ is not a solution to the inequality.

Conclusion

In this article, we have discussed how to solve inequalities, including strict and non-strict inequalities, and how to write equivalent inequalities. We have also provided example solutions to illustrate the concept.