Problem 43 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ 5 x-7 x \leq x+2 $$

Step-by-Step Solution

Answer

The solution is \( \{ x \,|\, x \geq -\frac{2}{3} \} \).

Step 1: Simplify the Expression

First, combine like terms on the left-hand side of the inequality. The terms \(5x\) and \(-7x\) combine to give \(-2x\). The inequality simplifies to: \[-2x \leq x + 2\]

Step 2: Move Variable to One Side

Subtract \(x\) from both sides to get all the \(x\) terms on one side of the inequality. This becomes: \[-2x - x \leq 2\] Thus, simplify to: \[-3x \leq 2\]

Step 3: Solve for x

To solve for \(x\), divide both sides of the inequality by \(-3\). Remember that when you divide by a negative number, you must flip the inequality sign: \[x \geq -\frac{2}{3}\]

Step 4: Write Solution in Set Notation

Express the solution in set notation. The set notation is based on the condition found: \[\{ x \,|\, x \geq -\frac{2}{3} \}\]

Key Concepts

Solution Set Notation

In mathematics, especially when dealing with inequalities, expressing the solution in an understandable format is crucial. This is where solution set notation comes into play. Solution set notation is a way of writing the answer to an equation or inequality that shows all possible values that satisfy the given condition. For example, if you solve an inequality and determine that all values of \(x\) greater than or equal to some number are solutions, you can use set notation to express this neatly.

Consider our solved inequality: \(x \geq -\frac{2}{3}\). In solution set notation, this is written as \( \{ x \,|\, x \geq -\frac{2}{3} \} \). This read as: "the set of all \(x\) such that \(x\) is greater than or equal to \(-\frac{2}{3}\)."

Using this notation:

• Curly brackets \( \{ \} \) indicate that a set is defined.
• The vertical bar "\(|\)" represents "such that" or "where."

This format is not only concise but also communicates the range of possible solutions effectively. Understanding this helps in interpreting the results of inequality problems and aids in representing and manipulating solution sets in a structured way.

Combining Like Terms

When you have multiple terms that involve the same variables, you can "combine" these terms to simplify your expression. This is incredibly important in solving equations and inequalities because it can significantly reduce complexity.

Like terms are terms that have the exact same variable raised to the same power. In our problem, \(5x\) and \(-7x\) are like terms because they both contain the variable \(x\) raised to the first power.

• Combine \(5x\) and \(-7x\) by summing their coefficients: \(5 + (-7) = -2\).
• The combined term becomes \(-2x\).

By combining like terms, the original expression \(5x - 7x \leq x + 2\) is simplified to \(-2x \leq x + 2\).

This simplification is vital because it makes it easier to perform further algebraic operations. Simplifying expressions not only reduces errors but also lays a clear path to solving equations and inequalities effectively.

Solving Linear Inequalities

A linear inequality is similar to a linear equation, but instead of having an equal sign, it uses inequality symbols like \(>\), \(<\), \(\geq\), or \(\leq\). Solving these inequalities involves finding a range of values, rather than a single solution, that makes the inequality true.

To solve the inequality \(-2x \leq x + 2\), follow these steps:

• First, get all \(x\) terms on one side: Subtract \(x\) from both sides to get \(-3x \leq 2\).
• Then solve for \(x\) by dividing both sides by \(-3\). Remember to flip the inequality sign because you're dividing by a negative number. This changes the inequality to \(x \geq -\frac{2}{3}\).

It's important to always remember:

• Flipping the inequality sign is crucial when multiplying or dividing both sides of an inequality by a negative number.
• Check your solution by testing values to ensure they satisfy the original inequality.

By understanding these steps, you can confidently tackle a wide range of inequality problems, always remembering the special rules like flipping the inequality sign when necessary.