Problem 48 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ 5(x-2) \leq 3(2 x-1) $$

Step-by-Step Solution

Answer

\[ \{ x \mid x \geq -7 \} \]

Step 1: Expand Both Sides

First, expand both sides of the inequality. On the left side, distribute the 5, so it becomes: \[ 5(x - 2) = 5x - 10 \]On the right side, distribute the 3, so it becomes:\[ 3(2x - 1) = 6x - 3 \]Now the inequality is:\[ 5x - 10 \leq 6x - 3 \]

Step 2: Isolate Variable Terms

Next, get all the terms containing the variable \(x\) on one side. Subtract \(5x\) from both sides:\[ 5x - 10 - 5x \leq 6x - 3 - 5x \]This simplifies to:\[ -10 \leq x - 3 \]

Step 3: Solve for the Variable

Now, solve for \(x\) by adding 3 to both sides of the inequality:\[ -10 + 3 \leq x - 3 + 3 \]This simplification results in:\[ -7 \leq x \]

Step 4: Write the Solution in Set Notation

Finally, express the solution in solution set notation. Since \(-7\leq x\), the solution set is:\[ \{ x \mid x \geq -7 \} \]

Key Concepts

Solution Set Notation

In algebra, solution set notation is a way to express the set of all solutions that satisfy a given inequality or equation. When solving inequalities, we use this notation to clearly represent the range of values that make the inequality true. In our example, after working out the inequality, we found that \(-7 \leq x\).

This means any number greater than or equal to -7 satisfies the inequality. The solution set notation to express this is:\[ \{ x \mid x \geq -7 \} \]

• \( \{ x \mid \ldots \} \) means "the set of all \( x \) such that".
• The condition "\( x \geq -7 \)" defines the values that belong to this set.

Solution set notation is useful in clearly communicating the set of values that are solutions to the inequality. Instead of using a verbal description or a number line, it provides a concise mathematical expression.

Expanding Expressions

Expanding expressions involves distributing a number across terms inside parentheses. This is an essential skill in algebra for simplifying and solving equations or inequalities. In the context of this problem, to solve the inequality \(5(x-2) \leq 3(2x-1)\), expanding both sides is the first step.

On the left side, we distributed 5 into \(x-2\):

• Multiply 5 by \(x\): \(5 \cdot x = 5x\).
• Multiply 5 by \(-2\): \(5 \cdot -2 = -10\).

So the left side becomes \(5x - 10\).On the right side, we distributed 3 into \(2x-1\):

• Multiply 3 by \(2x\): \(3 \cdot 2x = 6x\).
• Multiply 3 by \(-1\): \(3 \cdot -1 = -3\).

Resulting in \(6x - 3\).

Expanding expressions like this makes it easier to manipulate equations and solve inequalities by clearing the parentheses, allowing us to focus on the variable terms.

Step-by-Step Solution

A step-by-step solution is invaluable because it guides us through solving complexities in algebra like inequalities, one stage at a time. Here's how we approached solving our inequality:

1. **Expand Both Sides:** We began by expanding the expressions on both sides of the inequality \(5(x-2) \leq 3(2x-1)\) to get \(5x - 10\) and \(6x - 3\). These expansions simplify further operations.2. **Isolate the Variable Terms:** The next step is placing all terms with the variable on one side and constants on the other. Subtract \(5x\) from both sides: \[-10 \leq x - 3\].3. **Solve for the Variable:** Add 3 to both sides to isolate \(x\): \[-10 + 3 \leq x\] simplifies to \(-7 \leq x\).4. **Solution Set Notation:** Finally, express the solution in set notation as \(\{ x \mid x \geq -7 \}\).

This approach breaks down each part of the process, making it easier to follow and understand. Each step builds on the previous one, helping students see the logical progression in solving inequalities.