Solve the Inequality for k 6(k-11)> 12

Solution:

Step 1

Divide every term in the inequality $6(k-11)>12$ by $6$ to simplify the expression.

Step 1.1

Perform the division: $\frac{6(k-11)}{6}>\frac{12}{6}$.

Step 1.2

Simplify the left-hand side of the inequality.

Step 1.2.1

Eliminate the common factor of $6$.

Step 1.2.1.1

Remove the common factor: $\frac{\cancel{6}(k-11)}{\cancel{6}}>\frac{12}{6}$.

Step 1.2.1.2

Reduce $k-11$ by $1$: $k-11>\frac{12}{6}$.

Step 1.3

Simplify the right-hand side of the inequality.

Step 1.3.1

Compute the division of $12$ by $6$: $k-11>2$.

Step 2

Isolate the variable $k$ by moving all other terms to the opposite side of the inequality.

Step 2.1

Add $11$ to both sides of the inequality: $k>2+11$.

Step 2.2

Combine the constants: $k>13$.

Step 3

Express the solution in various forms.

Inequality Form: $k>13$ Interval Notation: $(13,\mathrm{\infty })$

Knowledge Notes:

The problem involves solving an inequality, which is a mathematical statement that relates two expressions with an inequality sign. Inequalities can be solved using similar methods to equations but require special consideration when multiplying or dividing by negative numbers, which reverses the inequality sign.

The steps taken to solve the inequality $6(k-11)>12$ are as follows:

1. Division: To simplify the inequality, we divide each term by the same non-zero number. This does not change the direction of the inequality as long as the number is positive.

2. Simplification: After dividing, we simplify the expression by cancelling out common factors and performing any arithmetic operations required.

3. Isolation of the Variable: We rearrange the inequality to isolate the variable on one side. This typically involves adding or subtracting terms on both sides of the inequality.

4. Solution Representation: The solution to an inequality can be represented in several ways, including inequality notation (e.g., $k>13$) and interval notation (e.g., $(13,\mathrm{\infty })$), which indicates the set of all numbers greater than 13.

When dealing with inequalities, it is crucial to remember that if we multiply or divide by a negative number, we must reverse the inequality sign. However, this situation does not arise in this particular problem since all operations involve positive numbers.