Problem 56 - Prealgebra

Question

Solve each inequality. $$2 x+5<17$$

Step-by-Step Solution

Answer

The solution is \(x < 6\).

Step 1: Isolate the Variable Term

To solve the inequality \(2x + 5 < 17\), our goal is to isolate the term with the variable. First, remove the constant term on the left side by subtracting 5 from both sides: \(2x + 5 - 5 < 17 - 5\). Simplifying both sides gives us \(2x < 12\).

Step 2: Solve for the Variable

Now, divide both sides of the inequality by 2 to solve for \(x\). This gives \(\frac{2x}{2} < \frac{12}{2}\). Simplifying gives the result \(x < 6\).

Key Concepts

Prealgebra

Prealgebra is a critical stepping stone in math education. It often includes fundamental math concepts that prepare students for algebra. In prealgebra, you'll explore basic operations, fractions, and the beginnings of algebraic thinking.

One of the primary goals in prealgebra is to understand variables and expressions. Variables are symbols used to represent numbers whose values can change. You'll often see them in equations and inequalities, paving the way for more complex algebraic concepts later.

Another essential part of prealgebra is becoming comfortable with mathematical operations as you go from concrete arithmetic calculations to more abstract reasoning. For instance, transitioning from seeing numbers as fixed values to understanding how they can move and change in equations.

• Builds foundation for algebra
• Introduces variables and expressions
• Focuses on abstract reasoning

Solving Inequalities

Understanding how to solve inequalities is a key component of algebra. Inequalities are similar to equations, but instead of showing equality, they show that one side is less than, greater than, or not equal to the other.

In the inequality given, we had: \(2x + 5 < 17\). The objective was to isolate \(x\) to discover the solution set. This typically involves inverse operations:

• Subtract any constants from both sides
• Divide or multiply to solve for the variable

Throughout, it's crucial to maintain the inequality's balance. A significant rule to remember is that when multiplying or dividing by a negative number, the inequality sign flips. Here, there's no need to flip the sign since we divided by a positive number.

Solving inequalities often requires you to visualize solutions on a number line, providing a clear picture of the solution set. It illustrates values that satisfy the inequality.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operation symbols. They do not contain equality or inequality signs, which distinguishes them from equations and inequalities.

In our inequality \(2x + 5 < 17\), the expression is \(2x + 5\). Breaking it down:

• \(2x\) is a term consisting of the coefficient (2) and the variable (\(x\))
• 5 is a constant term

Understanding each component of an expression is vital as it allows us to manipulate the expressions appropriately when solving equations or inequalities.

Operations on algebraic expressions include addition, subtraction, multiplication, and division. In solving the inequality, we manipulated the expression by subtracting terms and dividing by coefficients to isolate the variable.

Grasping algebraic expressions early on helps build problem-solving skills and algebraic thinking necessary for advanced math topics. The key takeaway is recognizing how each term interacts within the expression to solve algebraic problems.