Problem 72 - Algebra for College Students

Question

Give a step-by-step description of how you would solve the inequality \(-3>5-2 x\).

Step-by-Step Solution

Answer

The solution to the inequality is \(x > 4\).

Step 1: Isolate the Variable Term

First, we need to get the term with the variable on one side of the inequality. We start by subtracting 5 from both sides to eliminate the constant on the right side. This subtraction gives us: \[-3 - 5 > -2x\]Simplifying the left side, we have:\[-8 > -2x\].

Step 2: Solve for x

Next, we need to get \(x\) by itself. To do this, divide each side of the inequality by \(-2\). Remember, when you divide or multiply both sides of an inequality by a negative number, the inequality sign reverses direction.\[\frac{-8}{-2} < x\]Simplifying, we find:\[4 < x\].

Step 3: Write the Final Solution

The inequality solved shows that \(x\) has to be greater than 4. We can write the solution as:\[x > 4\].

Key Concepts

Algebraic Expressions

Algebraic expressions can seem a bit tricky at first, but they become manageable once you get the hang of them. An algebraic expression consists of variables, numbers, and operations. These can include addition, subtraction, multiplication, and division.
In the given inequality,

• -3 is a constant.
• 5 is also a constant.
• -2x is a term which includes a coefficient, -2, and a variable, x.

Algebraic expressions are used in inequations to compare values and determine relationships between them. Understanding these components is fundamental as they form the basis of how we solve inequalities, like \(-3 > 5 - 2x\). You'll often manipulate them to isolate variables and ultimately solve for unknown values.

Inequality Solutions

Solving inequalities usually involves finding out which values satisfy the conditions given by a particular inequality. With inequalities, you're not looking for just one specific answer, but rather a range of possible solutions.

• Consider expressions such as \(-3 > 5 - 2x\).
• The goal is to find all values of \(x\) that make this inequality true.

Here are key points to remember when solving inequalities:

• Perform similar operations on both sides of the inequality as you would with equations.
• Keep in mind how reversing the inequality works. Specifically, when you multiply or divide by a negative number, the inequality symbol must be flipped.

These properties help us identify the solution safely, ensuring we find the accurate range in which \(x\) satisfies the inequality.

Variable Isolation

The process of variable isolation is crucial for solving inequalities like \(-3 > 5 - 2x\). The goal is to rearrange the inequality so that the variable is alone on one side. This allows you to easily understand what values the variable can take to satisfy the inequality.

• Start by removing any terms not containing the variable from the same side as the variable.
• For example, in the step \(-3 - 5 > -2x\), we isolate terms with x by first moving the constant 5 across the inequality sign.
• Then, address the coefficient of x (here, -2). Divide the entire inequality by -2 to simplify.

Remember, when dividing by a negative number, flip the inequality sign. Hence, \(-8 > -2x\) transforms into \(4 < x\).
Variable isolation is an important skill that helps us solve inequalities methodically and accurately, ensuring that we clearly detail the range of solutions.