Problem 90 - Introductory Algebra for College Students

Question

Solve each inequality. \(6 x-3 \leq 3(x-1)\)

Step-by-Step Solution

Answer

\(x \leq 1\)

Step 1: Distribute on the right side

Begin by distributing the 3 on the right side of the inequality. This will result in:\(6x - 3 \leq 3x - 3\)

Step 2: Isolate x terms

Next, move all terms involving x to one side and the constants to the other side. This can be done by subtracting 3x from both sides, which results in:\(3x - 3 \leq 0\)

Step 3: Solving for x

Finally, solve the resulting equation for x by adding 3 to both sides, which gives:\(3x \leq 3\) Then divide both sides by 3 to solve for x:\(x \leq 1\)

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that simplifies expressions. It allows you to distribute, or "pass through," a multiplication operation over addition or subtraction within parentheses. This principle can be expressed as:

• For any numbers or variables, \[ a(b+c) = ab + ac \]

In the context of the inequality \( 6x-3 \leq 3(x-1) \), the distributive property is applied to the expression on the right side. Here, we distribute the 3 across the terms inside the parentheses \((x-1)\).
This gives:

• \[ 3 \cdot x - 3 \cdot 1 = 3x - 3 \]

Thus, the inequality becomes \( 6x - 3 \leq 3x - 3 \).
Understanding and applying the distributive property correctly is crucial as it sets up the expression for further solving.
It simplifies complex expressions and prepares them for the next steps in solving equations and inequalities.

Isolation of Variables

When solving inequalities, one of the most important steps is to isolate the variable on one side of the inequality. This means to move all terms with the unknown variable to one side and all constant terms to the other side.
To achieve this in our inequality example, we need to move the \(3x\) from the right side to the left.

• Start by subtracting \(3x\) from both sides: \[ 6x - 3x - 3 \leq 3x - 3x - 3 \].
• This simplifies to \[ 3x - 3 \leq 0 \].

By completing these steps, our inequality is now in a form where the variable is isolated on one side, ready for further solving.
Isolation allows us to see the inequality more clearly and is the key to solving for the variable.

Inequality Solving Steps

Solving inequalities follows similar steps to solving equations, with special attention to the inequality sign. Here’s a brief guide using our example:

• Start by simplifying both sides of the inequality using the distributive property if necessary.
• Move terms involving the variable to one side, as done in the isolation step.
• Simplify further by adding or subtracting constants:For instance, from \(3x - 3 \leq 0\), add 3 to both sides.
• This results in: \[3x \leq 3\].
• Finally, divide by the coefficient of the variable to solve for the variable itself:Divide both sides by 3 to get \[x \leq 1\].

Always remember to flip the inequality sign when multiplying or dividing by a negative number, although not applicable in this particular problem.
These steps provide a straightforward method to tackle any linear inequality.