Problem 17 - Algebra 2 and Trigonometry

Question

In \(13-22,\) solve each equation or inequality. Each solution is an integer. $$ 2 x+3 < x+15 $$

Step-by-Step Solution

Answer

The integer solutions are all integers less than 12.

Step 1: Understand the Inequality

We need to solve the inequality \(2x + 3 < x + 15\) to find the integer solutions for \(x\).

Step 2: Subtract \(x\) from Both Sides

Subtract \(x\) from both sides of the inequality to start isolating \(x\) on one side:\[2x + 3 - x < x + 15 - x\] which simplifies to:\[x + 3 < 15\]

Step 3: Subtract 3 from Both Sides

Subtract 3 from both sides to further isolate \(x\):\[x + 3 - 3 < 15 - 3\] which simplifies to:\[x < 12\]

Step 4: Interpret the Solution

The solution to the inequality is \(x < 12\), meaning any integer less than 12 satisfies the inequality. Therefore, the possible integer solutions are all integers \(x\) such that \(x < 12\).

Key Concepts

Integer Solutions

Integer solutions are values that satisfy an equation or inequality and are whole numbers. When solving inequalities, we often seek integer solutions that make the statement true. To find these solutions, we first solve the inequality using algebraic methods and then identify which integers fit the solution's condition.In our example, after solving the inequality, we identified that all integers less than 12 satisfy the inequality. This means the integer solutions for the inequality \(2x + 3 < x + 15\) are the set:

• \(..., -2, -1, 0, 1, 2, ...10, 11\)

These integers ensure the inequality holds true and illustrate the idea of integer solutions.

Inequality Manipulation

Inequality manipulation involves changing an inequality into a simpler form that is easier to solve. This process is much like solving regular equations, but with a few key differences to remember.To start, inequalities use signs like \(<, \leq, >, \geq\) instead of the equals sign. An important rule is that if you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.In our solution, we didn’t need to multiply or divide by negatives, so the steps were straightforward:

• We subtracted \(x\) from both sides to get \(x + 3 < 15\).
• Then, we subtracted 3 from both sides to yield \(x < 12\).

These steps illustrate how manipulation can lead to a simple inequality, allowing us to easily determine solutions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, letters (representing variables), and operation symbols. They play a vital role in forming equations and inequalities, acting as the foundation of algebra.In the inequality \(2x + 3 < x + 15\), we have the algebraic expression \(2x + 3\) on the left and \(x + 15\) on the right. Solving this inequality involved simplifying these expressions step by step:

• First, we subtracted \(x\) from both sides to simplify to \(x + 3 < 15\). This combined like terms, reducing the complexity.
• Next, we subtracted 3 from both sides, yielding the expression \(x < 12\).

Understanding the structure of these expressions is crucial, as it helps identify the steps needed to isolate variables and solve inequalities.