Solve the Inequality for x (x-4)/6> =(x-2)/9+5/18

Solution:

Step 1: Scale up the inequality by multiplying by 18 to eliminate fractions.

$$18\cdot \frac{x-4}{6}\ge 18\cdot (\frac{x-2}{9}+\frac{5}{18})$$

Step 2: Break down the simplification process.

Step 2.1: Focus on the left-hand side first.

Step 2.1.1: Identify and remove common factors.

$$\frac{18}{6}\cdot (x-4)\ge 18\cdot (\frac{x-2}{9}+\frac{5}{18})$$

Step 2.1.2: Express the simplified left-hand side.

$$3(x-4)\ge 18\cdot (\frac{x-2}{9}+\frac{5}{18})$$

Step 2.2: Now, simplify the right-hand side.

Step 2.2.1: Work on the term inside the parentheses.

Step 2.2.1.1: Adjust the fraction to have a common denominator.

$$3(x-4)\ge (\frac{2(x-2)}{18}+\frac{5}{18})\cdot 18$$

Step 2.2.1.2: Combine the terms over a common denominator.

$$3(x-4)\ge \frac{2(x-2)+5}{18}\cdot 18$$

Step 2.2.1.3: Simplify the combined terms.

$$3(x-4)\ge 2x-4+5$$

Step 2.2.1.4: Finalize the right-hand side.

$$3(x-4)\ge 2x+1$$

Step 3: Isolate the variable x.

Step 3.1: Expand the left-hand side.

$$3x-12\ge 2x+1$$

Step 3.2: Shift the x terms to one side.

$$3x-2x\ge 1+12$$

Step 3.3: Solve for x.

$$x\ge 13$$

Step 4: Express the solution in different forms.

Inequality Form: $$x\ge 13$$ Interval Notation: $$[13,\mathrm{\infty })$$

Knowledge Notes:

To solve a linear inequality similar to the one given, we follow these steps:

1. Eliminate Fractions: Multiply through by the least common multiple (LCM) of the denominators to clear fractions.

2. Simplify Each Side: Perform algebraic operations to simplify each side of the inequality independently.

3. Combine Like Terms: If there are like terms on the same side of the inequality, combine them.

4. Isolate the Variable: Get all terms with the variable on one side and constants on the other.

5. Solve the Inequality: Perform the operations needed to solve for the variable.

6. Check the Solution: Verify that the solution makes the original inequality true.

7. Express the Solution: Write the solution in inequality form and interval notation.

When solving inequalities, remember that multiplying or dividing both sides by a negative number reverses the inequality sign. However, this situation does not arise in this particular problem.