Question

Solve \(3 k-7 \geq 17\) A. \(k \geq 8\) B. \(k \leq 8\) C. \(k>\frac{10}{3}\) D. \(k \geq 24\)

Short answer

A. \(k \geq 8\)

Step-by-step solution

Add 7 to both sides

To isolate the term with the variable, add 7 to both sides of the inequality:\[3k - 7 + 7 \geq 17 + 7\]This simplifies to:\[3k \geq 24\]

Divide both sides by 3

Next, divide both sides of the inequality by 3 to solve for \(k\):\[\frac{3k}{3} \geq \frac{24}{3}\]This simplifies to:\[k \geq 8\]

Choose the correct option

The inequality \(k \geq 8\) matches option A. So, the correct answer is:A. \(k \geq 8\)

Key concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and rules for manipulating those symbols. In algebra, letters and other symbols are used to represent numbers and quantities in formulas and equations. This allows us to generalize arithmetic and solve a wide range of problems.

The exercise given is an algebraic inequality, which involves terms with a variable (in this case, 'k') and requires us to find the range of values that satisfy the inequality. Understanding basic algebraic operations such as addition, subtraction, multiplication, and division is crucial for solving these types of problems.

Variable Isolation

Variable isolation is a fundamental technique in algebra. It involves rearranging an equation or inequality so that the variable is on one side and everything else is on the other. This helps us to solve for the variable directly.

In our example, the inequality given was: \[3k-7 \geq 17\]The goal is to isolate 'k'.

Step 1: We start by removing the constant term from the side with the variable. Here, we added 7 to both sides to cancel out the -7:\[3k-7 + 7 \geq 17 + 7\]which simplifies to:\[3k \geq 24\]

Step 2: To further isolate 'k', we divide both sides by 3:\[\frac{3k}{3} \geq \frac{24}{3}\]This simplifies to:\[k \geq 8\]

This technique of isolating the variable makes it easier to see the relationship between 'k' and any numbers or constants in the inequality.

Inequality Solving

Solving inequalities is similar to solving equations but with some additional rules. Inequalities tell us about the relative size of two expressions and use symbols like >, <, ≥, and ≤.

When solving an inequality, we perform the same operations on both sides just as we do with equations. However, it's essential to remember that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.

In our example, we didn't deal with negative numbers, so the process followed the same steps as solving a regular equation. The solution to the inequality:\[k \geq 8\]means 'k' can be any number greater than or equal to 8. This is indicated by the '≥' symbol.

Lastly, always check the possible answers and verify the solution fits the original inequality. Here, option A (\[k \geq 8\]) matches our result perfectly.

Understanding how to solve inequalities is a vital skill in algebra. It allows us to determine the ranges and conditions where certain mathematical statements hold true.