Question

What is the solution of \(45 \geq 7 t+10 \geq 3\) ? (A) \(-5 \geq t \geq 1\) (B) \(5 \geq t \geq-1\) (C) \(5 \leq t \leq-1\) (D) \(-1 \geq t \geq 5\)

Short answer

The solution is (B) \(5 \geq t \geq -1\).

Step-by-step solution

Understand the Problem

The problem involves solving the compound inequality \(45 \geq 7t + 10 \geq 3\). We will solve the inequality in two parts: \(45 \geq 7t + 10\) and \(7t + 10 \geq 3\).

Solve the First Inequality

Start with the inequality \(45 \geq 7t + 10\). Subtract 10 from both sides to isolate the term with \(t\): \[ 45 - 10 \geq 7t \] This simplifies to \[ 35 \geq 7t \]. Divide both sides by 7 to solve for \(t\): \[ 5 \geq t \].

Solve the Second Inequality

Now solve the second inequality \(7t + 10 \geq 3\). Subtract 10 from both sides:\[ 7t \geq 3 - 10 \] This simplifies to \[ 7t \geq -7 \]. Divide both sides by 7 to solve for \(t\): \[ t \geq -1 \].

Combine Solutions

Combine both solutions from Steps 2 and 3. From \(5 \geq t\) and \(t \geq -1\), we get the compound inequality: \[ 5 \geq t \geq -1 \].

Match to Multiple Choice Answer

Compare the inequality \(5 \geq t \geq -1\) to the given options. The correct choice is (B) since it matches our solution: \(5 \geq t \geq -1\).

Key concepts

Compound Inequality

A compound inequality is a mathematical statement that relates two inequalities containing a common variable. In a compound inequality, you'll often find two separate conditions linked with the word "and" or "or." In our original problem, the compound inequality is given as \(45 \geq 7t + 10 \geq 3\). Here, the statement expresses two conditions:

• The expression \(7t + 10\) must be less than or equal to 45.
• It must also be greater than or equal to 3.

Compound inequalities are solved by handling each part separately and then seeing where these solutions overlap. This ensures that the solution satisfies both conditions simultaneously.

Solving Inequalities

Solving inequalities involves finding all values of the unknown variable that make the inequality true. Let's break down the steps for solving our compound inequality:

• Isolate the variable: Break the inequality into two parts and solve each for \(t\). For instance, solve \(45 \geq 7t + 10\) by subtracting 10 from both sides and then dividing by 7, resulting in \(5 \geq t\).
• Repeat for the second inequality: Similarly, for \(7t + 10 \geq 3\), subtract 10 and divide by 7 to get \(t \geq -1\).

By following these steps, you narrow down the possible values of \(t\). The result is that \(t\) satisfies the conditions set by both inequalities. Finally, by combining the results from both parts, you arrive at the solution of the compound inequality.

Mathematical Reasoning

Mathematical reasoning is a crucial part of solving inequalities and involves understanding relationships between numbers and operations. When tackling inequalities:

• Understand the variable positions: Recognize that the inequalities must cover the entire range where \(t\) satisfies both conditions.
• Ensure logical consistency: Check that each step follows logically from the last and that operations performed are valid across the inequality signs. For instance, dividing by a positive number retains the direction of inequality signs.
• Double-check work: This involves revisiting initial assumptions and ensuring solutions fit the original compound inequality.

Good mathematical reasoning doesn't just help in arriving at the solution but also in understanding the process and making sure it's accurate.

Algebra

Algebra is at the heart of solving inequalities. It provides the framework and tools needed for manipulating mathematical expressions involving variables. Let's see how algebra applies to our inequality:

• Using operations: Solving \(45 \geq 7t + 10\) involves subtraction and division, which are basic algebraic operations. These operations help isolate \(t\) and find its possible values.
• Combining results: Algebra also helps us combine results from multiple inequalities. For instance, combining \(5 \geq t\) and \(t \geq -1\) gives a comprehensive solution \(5 \geq t \geq -1\).
• Mathematical structure: Algebra teaches us how to maintain structure in our solutions. In this context, ensuring that any modifications to inequalities are systematically applied.

Mastering algebra allows for smooth resolution of not just inequalities, but various mathematical problems involving equations and expressions.