Problem 10 - Thomas Calculus

Question

In Exercises \(5-12,\) solve the inequalities and show the solution sets on the real line. $$ \frac{6-x}{4}<\frac{3 x-4}{2} $$

Step-by-Step Solution

Answer

The solution to the inequality is \(x > 2\), represented on the number line as an open interval starting from 2 and extending to the right indefinitely.

Step 1: Clear Fractions by Multiplying by 4

To clear fractions, multiply every term by the least common denominator, which is 4 in this case. This gives us:\[ (6-x) < 2(3x - 4) \]

Step 2: Distribute and Simplify

Distribute on the right side of the inequality:\[ 6 - x < 6x - 8 \]

Step 3: Add x to Both Sides

Add \(x\) to both sides to combine like terms and move \(x\) to one side:\[ 6 < 6x - 8 + x \] which simplifies to \[ 6 < 7x - 8 \]

Step 4: Add 8 to Both Sides

To isolate terms involving \(x\), add 8 to both sides of the inequality:\[ 6 + 8 < 7x \] which results in \[ 14 < 7x \]

Step 5: Divide Both Sides by 7

To solve for \(x\), divide both sides by 7:\[ x > 2 \]

Step 6: Solution Set and Graph on the Real Line

The solution set is \(x > 2\). On a number line, this is represented by an open interval starting from 2, and extending to the right infinitely. The point at 2 is open, indicating that it is not included in the solution.

Key Concepts

Real Number Line

The real number line is a visual representation of all real numbers. It extends infinitely in both directions, but for solving inequalities, we often focus on a specific portion.
Each point on the line corresponds to exactly one real number. When we depict solutions to inequalities, we use segments or rays on this line.
For instance, if a solution to an inequality is given as \(x > 2\), it means we consider all numbers greater than 2. These numbers are represented by a ray starting just after 2, going towards positive infinity. The point at 2 is not included, so it is marked with an open circle.

Fractions in Inequalities

Fractions in inequalities can sometimes make the equation look complex. The key is to simplify them to make the problem easier to solve.
One effective method is to eliminate fractions by multiplying each term by the least common denominator. This step is crucial as it keeps the inequality balanced while simplifying the terms.
For example, consider the inequality \(\frac{6-x}{4}<\frac{3 x-4}{2}\). To clear the fractions, multiply every term by 4, the least common denominator. This makes it simpler to handle:

• The new inequality is \(6-x<2(3x-4)\).
• This step retains the same relationships between numbers but in a more manageable form.

Inequality Solutions

Solving inequalities is about finding all values of the variable that make the inequality true. It involves several steps and logical operations.
In the process, after clearing fractions, simplify and rearrange terms to isolate the unknown variable on one side. Remember:

• Each action you take, such as adding, subtracting, or dividing, should be done to both sides of the inequality.
• Special care must be taken when multiplying or dividing by negative numbers, as this reverses the inequality sign.

Following these steps:

• Clear fractions and rearrange terms to isolate the variable.
• Solve the inequality, for instance here, we find \(x > 2\).

This tells us that any value greater than 2 satisfies the original inequality.

Graphical Solution Representation

Visualizing inequality solutions on the real number line helps understand them better. This graphical representation shows which values fit the inequality.
For the solution \(x > 2\), it's shown as:

• A ray starting just beyond the number 2, extending to the right.
• An open circle at 2 to indicate this number is not included in the solution.

This visual method clearly illustrates the concept and helps in understanding the solution set more concretely. It gives a clearer picture of all possible solutions to the inequality along the number line.