Problem 86 - Elementary Algebra

Question

Solve the inequalities. $$ -2 y \geq 14 $$

Step-by-Step Solution

Answer

Question: Solve the inequality -2y ≥ 14 and provide the solution. Answer: The solution to the given inequality is y ≤ -7.

Step 1: Identify the inequality

In this exercise, we are given the inequality: $$ -2y \geq 14 $$

Step 2: Isolate the variable y

In order to isolate y, we will divide both sides of the inequality by -2. Remember to reverse the inequality symbol when dividing by a negative number: $$ y \leq \dfrac{14}{-2} $$

Step 3: Simplify the inequality

Now, divide 14 by -2 to get the final solution: $$ y \leq -7 $$ The solution to the inequality is: $$ y \leq -7 $$

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a crucial skill in solving inequalities. It involves rearranging expressions and equations to isolate variables and find solutions. When dealing with inequalities, the process is similar to that of solving equations, with a few additional rules. To solve the inequality \(-2y \geq 14\), we start by performing operations that will isolate the variable \(y\). This might involve:

• Addition or subtraction of numbers from both sides.
• Multiplication or division of both sides by a number.

In this particular inequality, we need to divide both sides by \(-2\) to solve for \(y\). This operation is part of the algebraic manipulation that helps isolate the variable. Remember that your goal is to have the variable \(y\) on one side of the inequality, so you can clearly see its relationship with the number on the other side.

Negative Numbers

Negative numbers can bring an additional layer of complexity to solving inequalities. These numbers are less than zero and appear on the left side of a number line. When manipulating inequalities that involve negative numbers, a key point to remember is that dividing or multiplying both sides by a negative number will reverse the direction of the inequality symbol. For example, in the inequality \(-2y \geq 14\), dividing both sides by \(-2\) not only isolates the variable \(y\) but also changes the \(\geq\) to \(\leq\). This reversal is crucial and must be remembered each time you handle negative numbers in inequalities to ensure the solution remains valid.

Inequality Symbols

Understanding and correctly using inequality symbols are vital when working with inequalities. The primary symbols used in inequalities are:

• \(>\) : greater than
• \(<\) : less than
• \(\geq\) : greater than or equal to
• \(\leq\) : less than or equal to

In our exercise, the original inequality \(-2y \geq 14\) uses the \(\geq\) sign, indicating that \(-2y\) is greater than or equal to 14. After dividing both sides by \(-2\), the inequality sign flips to \(\leq\), making the solution \(y \leq -7\). The flipping of the symbol is an essential step in ensuring that the solution is accurate when you divide or multiply both sides by a negative number. It changes the direction of the inequality, reflecting the new mathematical relationship between the expressions on either side of the symbol.