Problem 15 - Algebra Form and Function

Question

Each of the inequalities can be solved by performing two operations on both sides. State the operations in order of use and solve the inequality. $$ 13 \leq 25-3 a $$

Step-by-Step Solution

Answer

Answer: The solution for the given inequality is \(a \leq 4\).

Step 1: Subtract 25 from both sides of the inequality

Subtract 25 from both sides to isolate the term with the variable \(a\). $$ 13 - 25 \leq 25 - 3a - 25 $$

Step 2: Simplify the inequality

Now, simplify the inequality by combining the numbers on both sides: $$ -12 \leq -3a $$

Step 3: Divide both sides by -3

To solve for \(a\), divide both sides of the inequality by -3. Since we’re dividing by a negative number, we need to reverse the inequality sign. $$ \frac{-12}{-3} \geq \frac{-3a}{-3} $$

Step 4: Simplify the inequality and find the solution

Now, simplify the inequality and find the solution for \(a\): $$ 4 \geq a $$ So, the solution for the inequality is \(a \leq 4\).

Key Concepts

Solving Inequalities

In mathematics, inequalities express a relationship where one number is not equal to another, often denoting if one number is greater or lesser than the other. Solving inequalities follows a process somewhat similar to solving equations. We look for the values that make the inequality true, allowing us to understand the range of possible solutions.

The step-by-step process usually involves simplifying the inequality to isolate the variable. Here's a general outline:

• Isolate the variable: Start by getting the variable term on one side of the inequality by adding, subtracting, multiplying, or dividing both sides.
• Simplify: Perform the necessary operations to simplify the numbers and terms.
• Solution Range: Determine the range of values that satisfy the inequality.

By following these steps, you'll find a range or a specific value that serves as the solution to the inequality.

Inequality Operations

While solving inequalities, certain operations can be applied to both sides, helping to isolate and solve for the desired variable. These operations include:

• Adding or subtracting: Similar to solving equations, you can add or subtract the same value from both sides of the inequality without changing its direction. For example, if you subtract 25 from both sides of the inequality \(13 \leq 25 - 3a\), you are left with \(-12 \leq -3a\).
• Multiplying or dividing: You can multiply or divide both sides by the same positive number without altering the inequality sign. However, particular care must be taken when multiplying or dividing by a negative number, as this requires reversing the inequality sign.

By understanding the operations you can apply, you can effectively solve inequalities and find solutions where it becomes essential to perform operations to balance both sides. Remember, the goal is to isolate the variable to determine its possible values.

Reversing Inequality Sign

One crucial aspect of inequalities is the need to reverse the inequality sign under specific conditions. This comes into play mostly in the context of multiplication or division by a negative number.

For instance, in the inequality \(-12 \leq -3a\), dividing both sides by the negative number -3 implies that the inequality sign must be reversed. Hence, it becomes \(4 \geq a\). This reversal is critical to maintaining the truth of the inequality.

• Why reverse the sign? Negative multiplication or division theoretically flips the entire number line, so a lower number becomes larger than a higher one and vice versa. This change represents a fundamental property that can't be overlooked when solving inequalities.

In summary, whenever multiplying or dividing both sides of an inequality by a negative number, always remember to reverse the inequality sign to maintain a correct and valid solution.