Problem 81 - Elementary Algebra

Question

Solve the inequalities. $$ y+3<15 $$

Step-by-Step Solution

Answer

Answer: The range of values for y that will satisfy the inequality is \(y<12\), which means any value of y less than 12 will satisfy the inequality.

Step 1: Write down the inequality

We are given the inequality: $$ y+3<15 $$

Step 2: Subtract 3 from both sides of the inequality

In order to isolate y, we need to subtract 3 from both sides of the inequality. This will give us: $$ y+3-3 < 15-3 $$

Step 3: Simplify the inequality

Now we simplify the inequality by performing the subtraction on both sides: $$ y < 12 $$

Step 4: State the solution

The solution to the inequality is \(y<12\). This means that any value of y that is less than 12 will satisfy the inequality.

Key Concepts

Solving Inequalities

When dealing with inequalities, the main goal is to find the range of values that satisfy the given condition. Inequalities are mathematical statements that relate expressions to one another using inequality symbols like \(<, >, \leq,\) and \( \geq\). For example, in the inequality \(y+3<15\), our task is to identify all possible values of \(y\) that make the inequality true. Unlike equations, solutions to inequalities are often ranges of numbers rather than single values. When solving, we use similar techniques as we do with equations, such as adding, subtracting, multiplying, or dividing both sides by the same number. We must remember that when multiplying or dividing by a negative number, the direction of the inequality sign is reversed.Here’s a quick tip: always double-check the direction of your inequality sign as you solve! It's crucial for ensuring the solution is accurate.

Step-by-step Solution

A step-by-step approach is the best way to tackle solving inequalities. It helps to systematically isolate the variable. Here's how it works:

• Write down the given inequality, so you have a clear starting point. In our example, this was \(y+3<15\).
• Determine what operation is needed to isolate the variable. Here, we needed to subtract 3 to get the variable by itself.
• Perform the same operation on both sides of the inequality. Once we subtracted 3 from both sides, we simplified the inequality to \(y<12\).
• Finally, state the resultant solution in a clear way. Here, any \(y\) less than 12 satisfies the original inequality.

Following these steps with care not only helps find the solution but builds understanding of the logical flow in solving inequalities.

Elementary Algebra

Elementary Algebra is the cornerstone of understanding how to manipulate mathematical expressions and solve equations and inequalities. It involves the use of symbols and variables to represent numbers and express relationships. Key principles of algebra include:

• Understanding variables as placeholders for unknown values. In our example, \(y\) is the variable.
• Executing arithmetic operations like addition, subtraction, multiplication, and division on both sides of an equation or inequality.
• Maintaining balance, meaning whatever operation you do to one side, you must do to the other to keep the expression equivalent.

Mastering these concepts helps in developing problem-solving skills that are applicable across different areas of mathematics! It's essential to approach these foundations with lots of practice and curiosity, as they form the basis for more advanced studies.