Problem 54 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ -2(x-4)-3 x<-(4 x+1)+2 x $$

Step-by-Step Solution

Answer

The solution set is \(\{x \mid x > 3\}\).

Step 1: Expand both sides of the inequality

First, expand the expressions on both sides of the inequality. On the left-hand side: \(-2(x-4) = -2x + 8\).Including the remaining expression, the left-hand side becomes \(-2x + 8 - 3x\).On the right-hand side: \(-(4x+1) = -4x - 1\).Including the remaining expression, the right-hand side is \(-4x - 1 + 2x\).

Step 2: Simplify both sides

Combine like terms on both sides of the inequality.Left-hand side: Combine \(-2x\) and \(-3x\) to get \(-5x + 8\).Right-hand side: Combine \(-4x\) and \(+2x\) to get \(-2x - 1\). The inequality now is:\(-5x + 8 < -2x - 1\).

Step 3: Isolate the variable term on one side

Add \(5x\) to both sides to get the variable on one side of the inequality:\(8 < 3x - 1\).

Step 4: Isolate the constant term on the other side

Add 1 to both sides to move the constant term:\(9 < 3x\).

Step 5: Solve for the variable

Divide both sides by 3 to completely isolate \(x\):\(3 < x\).In solution set notation, this is written as:\(\{x \mid x > 3\}\).

Key Concepts

Solution Set Notation

Solution set notation is a way of expressing the result of inequality solutions in a clear and concise manner. When we solve inequalities, we often find a range of values that satisfy the inequality rather than a single number. Solution set notation effectively communicates these ranges, making it easy for anyone to understand.

To write in solution set notation, we use curly braces with a variable inside, followed by a vertical bar that stands for "such that." For example, if we solve an inequality and find that the variable "x" is greater than 3, we write this solution as \( \{ x \mid x > 3 \} \). Here, \( \{ \} \) indicates the set, \( x \) is the variable, and \( > 3 \) describes the condition for the values that the variable can take.

Using solution set notation is especially helpful in mathematics because it is precise and universally understood. Whether you are dealing with inequalities or other mathematical problems, this notation offers clarity.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operators such as addition, subtraction, multiplication, and division. They form the building blocks of algebra and are used to represent real-world situations in mathematical terms.

In our exercise, we had the expression \( -2(x-4)-3x \). This is an example of an algebraic expression, which involves both numbers and variables. The key to solving equations and inequalities is to manipulate these expressions to find the values of the unknown variables.

There are several crucial operations that can be performed on algebraic expressions, such as:

• Expanding: Distributing multiplication over addition to remove parentheses, e.g., expanding \(-2(x-4)\) to \(-2x + 8\).
• Simplifying: Combining like terms to reduce the expression to its simplest form, such as combining \(-2x\) and \(-3x\) to get \(-5x\).

Understanding these operations makes it easier to solve more complex problems and develop a deeper understanding of algebra.

Solving Linear Inequalities

Solving linear inequalities is very similar to solving linear equations, with a few additional considerations. The goal is to isolate the variable on one side of the inequality, just like in an equation, but we have to consider the direction of the inequality sign.

A few key steps are usually involved:

• Expanding and Simplifying: As shown in the exercise, expand and simplify expressions on both sides. Combine like terms to make the inequality more manageable.
• Isolating the Variable: Use addition, subtraction, multiplication, or division to move terms around and get the variable alone on one side of the inequality, e.g., adding \(5x\) to both sides.
• Solving for the Variable: Once the variable is isolated, make sure to divide or multiply to solve for it, as shown when dividing by 3 to get \(x\) by itself.

An important thing to remember is when multiplying or dividing by a negative number, which is not the case in our specific example, the inequality sign must be reversed. Solving inequalities is a handy tool in many areas of mathematics and helps build the skills necessary for tackling more complex problems.