Problem 47 $$ \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 47

$$ \text { For Problems 23-60, solve each inequality. (Objective 1) } $$ $$ 3>4 x-2 $$

Short Answer

Expert verified

$ x < \frac{5}{4} $

Step by step solution

Add 2 to both sides

To isolate the variable term, we need to get rid of the constant on the right side. Add 2 to both sides of the inequality: \[3 + 2 > 4x - 2 + 2\] Simplifying both sides gives us:\[5 > 4x\]

Divide both sides by 4

To solve for $x$, divide both sides of the inequality by 4, which is the coefficient of $x$: \[\frac{5}{4} > x\] This simplifies to:\[ x < \frac{5}{4} \]

Conclusion

The inequality $ x < \frac{5}{4} $ shows the values that $x$ can take.

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Key Concepts

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

Inequality Symbols

Inequality symbols are similar to equal signs, but they show that two values are not exactly the same. Instead, they describe a range of possible values.
The most common inequality symbols include:

Greater than (>): This means the value on the left is larger than the one on the right.
Less than (<): This indicates the value on the left is smaller than the one on the right.
Greater than or equal to (鈮�): This asserts the left value is either larger than or equal to the right value.
Less than or equal to (鈮�): This states the left value is either smaller than or equal to the right value.

When solving inequalities, the objective is to find all values of the variable that make the inequality true.
In the expression given, "3 > 4x - 2", the solution process aims at isolating the variable to determine the range of values that satisfy the inequality.

Isolating Variables

Isolating variables is a fundamental step in solving inequalities.
It involves manipulating the inequality to leave the variable on one side, allowing us to see the possible values it can take. Let's go through this for the inequality given:

First, we need to eliminate any constants from the side containing the variable. In "3 > 4x - 2", we add 2 to both sides resulting in "5 > 4x". This step ensures only terms containing the variable remain on that side.
Next, we simplify by dealing with the coefficient of the variable. Here, divide both sides by 4 to achieve the format where the variable "x" is isolated: $\frac{5}{4} > x$.

Once the variable is isolated, it becomes clear which values satisfy the inequality, showing that any number less than $\frac{5}{4}$ is a potential solution.
Each manipulation step should preserve the inequality's balance by applying the same operation on both sides.

Step-by-step Solutions

Step-by-step solutions are crucial to understanding complex problems like inequalities, particularly for beginners.
They ensure that every small operation or correction is not missed, which can be vital for grasping basic math operations. In our exercise:

Step 1 involved simplifying the inequality by removing the constant on the variable鈥檚 side. Adding 2 to both sides gave the simpler inequality, "5 > 4x".
Step 2 required dividing by the coefficient of the variable. This separation made it easier to see the solution, represented as $x < \frac{5}{4}$.
Step 3 was simply stating the conclusion once the inequality was simplified, highlighting the range of values that work.

Breaking down complex problems into smaller, manageable pieces is the essence of step-by-step solutions.
This process not only makes solving easier but also helps in understanding the logic behind each mathematical operation.

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91影视

$$ \text { For Problems 23-60, solve each inequality. (Objective 1) } $$ $$ 3>4 x-2 $$

Short Answer

Step by step solution

Add 2 to both sides

Divide both sides by 4

Conclusion

Key Concepts

Inequality Symbols

Isolating Variables

Step-by-step Solutions

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