Problem 52 $$ \text { For Problems 23-60,... [FREE SOLUTION]

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Elementary Algebra

Jerome E. Kaufmann, Karen L. Schwitters

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 3: Problem 52

$$ \text { For Problems 23-60, solve each inequality. (Objective 1) } $$ $$ 36 \geq-7 t+1 $$

Short Answer

Expert verified

The solution is $ t \geq -5 $.

Step by step solution

Simplify the inequality

Start by simplifying and isolating the variable in the inequality. You have: \[36 \geq -7t + 1\] Subtract 1 from both sides to remove the constant on the right-hand side: \[36 - 1 \geq -7t\] which simplifies to: \[35 \geq -7t\]

Isolate the variable

To solve for $ t $, divide both sides of the inequality by $-7$. Remember, when you divide or multiply an inequality by a negative number, you must reverse the inequality sign. So, dividing by $-7$, we have: \[\frac{35}{-7} \leq t\] which simplifies to: \[-5 \leq t\]

Write the solution

You can express the solution in a standard form, noting that the inequality shows the values of $ t $ are greater than or equal to $-5$: \[t \geq -5\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate the Variable

When working with inequalities, such as the example $36 \geq -7t + 1$, the first step is to isolate the variable. This process involves rearranging the inequality so that we can easily see what values the variable, typically $t$, can take. The primary aim is to have the variable on one side of the inequality on its own.
Here are the steps you can follow to isolate:

First, identify and remove any constants from the side of the inequality where the variable resides. In this case, subtract $1$ from both sides.
This leads us to: $36 - 1 \geq -7t$.
Simplify this to $35 \geq -7t$.

This rearrangement ensures that you're moving steadily towards having the variable $t$ isolated on one side of the inequality. This incremental approach helps in understanding each part of the inequality better.

Reverse Inequality Sign

An important rule when solving inequalities is what happens if you multiply or divide both sides by a negative number. This step involves reversing the inequality sign. It might sound tricky, but it鈥檚 quite logical once you break it down.
Here's why and when you reverse the sign:

When you multiply or divide by a negative number, the direction of the inequality must switch. Think of it like flipping the sign on a number line.
In our example, we divide both sides of the inequality $35 \geq -7t$ by $-7$.
Because $-7$ is negative, the inequality sign flips, giving us $-5 \leq t$.

This flipping of the inequality is crucial in maintaining the truth of the statement represented by the inequality. Practice helps in remembering this rule.

Secondary Mathematics

The topic of solving inequalities is an essential part of secondary mathematics. It builds the foundation for more complex mathematical reasoning and problem-solving skills.
Understanding these concepts is crucial because:

Inequalities teach students to consider a range of solutions rather than a single answer. This is more reflective of real-world scenarios where variables can vary within limits.
Learning to manipulate inequalities enhances algebraic skills, strengthening students' capacity to solve complex equations soon.
It's a stepping stone into understanding functions, graphing, and even calculus later on.

Mastering secondary mathematics involving inequalities promotes logical thinking and application skills in broader mathematical contexts, ensuring students are well-prepared for more advanced studies.

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91影视

$$ \text { For Problems 23-60, solve each inequality. (Objective 1) } $$ $$ 36 \geq-7 t+1 $$

Short Answer

Step by step solution

Simplify the inequality

Isolate the variable

Write the solution

Key Concepts

Isolate the Variable

Reverse Inequality Sign

Secondary Mathematics

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