Problem 51 $$ \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 51

$$ \text { For Problems 23-60, solve each inequality. (Objective 1) } $$ $$ -38 \geq-9 t-2 $$

Short Answer

Expert verified

The solution is $t \geq 4$.

Step by step solution

Isolate the variable term

To solve the inequality $-38 \geq -9t - 2$,we first need to isolate the variable term. We can do this by adding 2 to both sides:\[-38 + 2 \geq -9t - 2 + 2\]This simplifies to:\[-36 \geq -9t\]

Solve for the variable

Now that the variable term is isolated, we need to solve for $t$. Divide both sides by $-9$ to solve for $t$:\[\frac{-36}{-9} \leq t\]Note: The direction of the inequality changes because we are dividing by a negative number. This simplifies to:\[4 \leq t \]

Express the solution

Since $4 \leq t$, it can also be expressed as:$t \geq 4$This means that the solution to the inequality is $t$ being any number greater than or equal to 4.

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Key Concepts

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

Isolation of Variables

When solving inequalities, one of the first and most crucial steps is the isolation of the variable you are solving for. Essentially, this means getting the variable by itself on one side of the equation or inequality sign. In our example, you have \[-38 \geq -9t - 2\]. Here, the term with the variable is $-9t$.
To isolate $t$, we aim to move all other numbers to the other side of the inequality. To do this:

Look at what is added or subtracted to your variable term. Here, $-2$ is subtracted.
Perform the opposite operation to both sides. So, we add 2 to both sides.

This converts the inequality into \[-36 \geq -9t\], with $-9t$ now isolated.
Remember that isolating the variable helps simplify the problem and sets the stage for solving the inequality.

Inequality Direction Change

An important feature of solving inequalities, especially involving multiplication or division by a negative number, is the change in the direction of the inequality. When you divide or multiply each side of an inequality by a negative number, the inequality sign flips direction.
In our example, we have \[-36 \geq -9t\], and we need to solve for $t$.

We divide both sides by $-9$ to isolate $t$.
The operation results in $\frac{-36}{-9} \leq t$.

Notice how the "$\geq$" becomes "$\leq$", indicating the reversal due to dividing by a negative.
This is a fundamental rule in algebra because reversing the direction of the inequality ensures the relation between the values remains true.

Inequality Solution Steps

The solution of an inequality involves several sequential steps that, when completed, provide the answer you're looking for. Let's break down the steps with a simple list.

Start by simplifying each side of the inequality if needed, ensuring no like terms are left undistributed.
Isolate the variable term on one side of the inequality, like we did moving from $-38 \geq -9t - 2$ to $-36 \geq -9t$.
Solve for the variable using division or multiplication, making sure to change the inequality direction if you multiply or divide by a negative number.
Finally, express the solution in a form that's clear and concise: $t \geq 4$, for instance, means $t$ can be any number from 4 onwards.

So, remember to follow each step one at a time, ensuring accuracy at every turn. These steps form the foundation of solving inequalities effectively.

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91影视

$$ \text { For Problems 23-60, solve each inequality. (Objective 1) } $$ $$ -38 \geq-9 t-2 $$

Short Answer

Step by step solution

Isolate the variable term

Solve for the variable

Express the solution

Key Concepts

Isolation of Variables

Inequality Direction Change

Inequality Solution Steps

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