Problem 13 Solve each inequality and give a... [FREE SOLUTION]

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Essentials of Geometry for College Students 2nd

Margaret L. Lial, Barbara A. Brown, Arnold R. Steffensen

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 5: Problem 13

Solve each inequality and give a reason for each step in the solution. $$1-3 x < 8$$

Short Answer

Expert verified

x > -\frac{7}{3}

Step by step solution

- Isolate the term with variable

First, subtract 1 from both sides of the inequality to isolate the term with the variable $-3x$: $1 - 3x - 1 < 8 - 1$

- Simplify

Simplify both sides: $-3x < 7$

- Divide by coefficient

Divide every term by $-3$ to solve for $x$. Don't forget to reverse the inequality symbol when dividing by a negative number: $x > \frac{7}{-3}$

- Simplify further

Simplify the fraction to get the final solution: $x > -\frac{7}{3}$

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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, the first step often involves isolating the variable. This means you want to get the variable by itself on one side of the inequality. In the given problem, we start with the inequality:
$1 - 3x < 8$
We need to isolate the term with the variable $-3x$. To do this, we subtract 1 from both sides:
$1 - 3x - 1 < 8 - 1$
This simplifies to:
$-3x < 7$
By isolating the variable, we make it easier to solve the inequality and understand what values the variable can take.

Inequality Properties

Inequality properties are essential in understanding how to solve inequalities correctly. These properties include:

Addition/Subtraction Property: If you add or subtract the same number from both sides of an inequality, the inequality remains true. In our example, subtracting 1 from both sides maintains the inequality $1 - 3x - 1 < 8 - 1$.
Multiplication/Division Property: If you multiply or divide both sides of an inequality by a positive number, the direction of the inequality remains the same. However, if you multiply or divide by a negative number, you must reverse the inequality symbol. This is crucial when solving for the variable, as shown in the next step when dividing by $-3$.

Remembering these properties ensures that you maintain the correct direction of the inequality as you solve it.

Reverse Inequality

A critical point to remember when working with inequalities is that dividing or multiplying both sides of an inequality by a negative number will reverse the direction of the inequality. In our example, we had:
$-3x < 7$
To solve for $x$, we divide both sides by $-3$. Since we are dividing by a negative number, we must reverse the inequality:
$x > \frac{7}{-3}$
This step alters the inequality symbol from $<$ to $>$, which is crucial for obtaining the correct solution set.

Simplifying Fractions

The final part of solving inequalities often involves simplifying fractions. Simplifying makes the answer more understandable. In our example:
$x > \frac{7}{-3}$
We simplify this to:
$x > -\frac{7}{3}$
Simplifying fractions involves reducing them to their simplest form or recognizing parts of the fraction that can be combined. This makes it easier to interpret the solution and ensures clarity in mathematical communication.

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91影视

Solve each inequality and give a reason for each step in the solution. $$1-3 x < 8$$

Short Answer

Step by step solution

- Isolate the term with variable

- Simplify

- Divide by coefficient

- Simplify further

Key Concepts

Isolation of Variables

Inequality Properties

Reverse Inequality

Simplifying Fractions

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