Problem 12 - The Calculus with Analytic Geometry

Question

$$ \text { In Exercises } 11 \text { through } 32 \text {, find the solution set of the given inequality and illustrate the solution on the real number } $$ $$ 3-x<5+3 x $$

Step-by-Step Solution

Answer

The solution set is \(x > -1/2\).

Step 1 - Move Variables to One Side

Start by moving all the terms containing the variable to one side of the inequality. Subtract \(3x\) from both sides:\(3 - x - 3x < 5 + 3x - 3x\)

Step 2 - Combine Like Terms

Combine the like terms on each side:\(3 - 4x < 5\)

Step 3 - Isolate the Variable

Subtract 3 from both sides to isolate the term that contains \(x\):\(3 - 4x - 3 < 5 - 3\)\(-4x < 2\)

Step 4 - Solve for the Variable

Divide both sides by -4, remembering to reverse the inequality sign when dividing by a negative number:\(-4x / -4 > 2 / -4\)\( x > -1 / 2\)

Step 5 - Illustrate the Solution

On a number line, draw an open circle at \(-1/2\) and shade all numbers to the right to indicate that \(x\) can be any value greater than \(-1/2\).

Key Concepts

Inequalities in Algebra

When we work with inequalities in algebra, our goal is to find a set of values that satisfy the given inequality. Inequalities are similar to equations, but instead of having an equal sign (\( = \)), they use less than (\( < \)), greater than (\( > \)), less than or equal to (\( \leq \)), and greater than or equal to (\( \geq \)) signs.
For example, in the inequality \( 3 - x < 5 + 3x \), we are looking for all the values of \( x \) that make this statement true.
The primary difference between solving equations and inequalities is how we handle the inequality signs, especially when multiplying or dividing by negatives.

Isolation of Variables

To solve inequalities, we often need to isolate the variable. This means getting the variable on one side of the inequality and the constant terms on the other.
In our exercise, we start by moving all \( x \) terms to one side: \( 3 - x - 3x < 5 + 3x - 3x \).
After simplifying, we get \( 3 - 4x < 5 \). Now, we need to isolate \( x \) by getting rid of the constants on the left side. We subtract 3 from both sides to get \( -4x < 2 \).
Isolation helps us to get a clearer view of the variable's relationship with the constant.

Reversing Inequality Signs

A critical point in solving inequalities is the need to reverse the inequality sign when multiplying or dividing by a negative number.
In our example, once we have \( -4x < 2 \), we must divide by -4 to solve for \( x \). This means that we need to reverse the inequality sign: \( -4x / -4 > 2 / -4 \).
Therefore, \( x > -1/2 \).
This rule is essential to remember because it ensures that the inequality's direction accurately represents the relationship between the numbers.

Number Line Representation

Once we have solved the inequality, it is useful to represent the solution on a number line.
In our case, \( x > -1/2 \) means that \( x \) can be any number greater than \( -1/2 \).
On the number line, we indicate this solution with an open circle at \( -1/2 \) (to show that \( -1/2 \) itself is not included) and shade all numbers to the right.
This visual representation helps to quickly see the range of solutions.