Problem 53 - Prealgebra

Question

Solve each inequality. $$4 y>24$$

Step-by-Step Solution

Answer

The solution is \(y > 6\).

Step 1: Understanding the Inequality

The inequality given is \(4y > 24\). We want to find all values of \(y\) that satisfy this inequality.

Step 2: Isolate the Variable y

To find the values of \(y\), we need to isolate \(y\) on one side of the inequality. Since \(y\) is multiplied by 4, we will divide both sides of the inequality by 4 to isolate \(y\).

Step 3: Divide Both Sides by 4

Divide each side of the inequality by 4: \[ \frac{4y}{4} > \frac{24}{4} \]. This simplifies to \(y > 6\).

Step 4: Verify the Solution

Any \(y\) greater than 6 will satisfy the original inequality \(4y > 24\). We can choose a number greater than 6, like 7, and substitute it back into the inequality to verify: \(4 imes 7 = 28\), which is greater than 24.

Key Concepts

Prealgebra

Prealgebra serves as the foundation for all higher-level math, providing you with the necessary skills to navigate through equations and inequalities. In prealgebra, we start with operations like addition, subtraction, multiplication, and division. These operations are used to solve basic equations and inequalities.
Understanding these fundamental operations helps in exploring number properties and variable manipulation. This will later be vital in algebra, where you will handle more abstract concepts and complex equations.
In the context of solving inequalities, prealgebra equips you with the ability to manipulate numbers and understand the relationships between them. It allows you to practice the logical thinking needed to find solutions where you aren't just calculating but reasoning through each step of your problem-solving process.

Isolation of Variables

The isolation of a variable in an equation or inequality means rearranging the equation so that the variable stands alone on one side. This is a fundamental concept in solving equations and inequalities.
When isolating a variable, you perform operations to both sides of the equation or inequality to keep it balanced. Think of it like a seesaw: if you add or remove weight from one side, you need to do the same to the other.
For the inequality \(4y > 24\), isolating the variable \(y\) required dividing both sides by 4. This operation eliminated the coefficient in front of \(y\), giving you \(y > 6\). Isolating variables helps in determining the solution set, showing clearly what range of values satisfies the equation or inequality.

Mathematical Inequalities

Mathematical inequalities are similar to equations, but instead of the sides being equal, one side is larger or smaller than the other. They explore the relationships between expressions and are often represented with symbols like \(>\), \(<\), \(\geq\), and \(\leq\).
Understanding inequalities involves more than simply finding a single solution; it requires determining a range of possible solutions. For example, with the inequality \(4y > 24\), the solution \(y > 6\) indicates that any number greater than 6 will satisfy that inequality.
Practitioners need to remember that any operation applied to an inequality must keep the inequality true, which involves careful thought when multiplying or dividing by negative numbers, since this would reverse the inequality symbol.

Division in Equations

Division is a crucial operation used when solving equations and inequalities, especially for isolating variables. When tackling \(4y > 24\), dividing both sides by the same value helped simplify to \(y > 6\).
While dividing in equations and inequalities, it's important always to apply the division rule to both sides equally to maintain the balance of the equation. This keeps the mathematical relationship unchanged while simplifying the terms.
In inequalities, particularly, you must be cautious. If dividing by a negative number, the inequality symbol must be flipped. Fortunately, in our example, since we divided by a positive number, the inequality direction remained the same. Always cross-verify your solutions by substituting values into the original inequality to ensure correctness.