Solving Linear Inequalities and Expressing Solutions in Interval Notation

Q1. Solve the following inequality. Graph the solution set and write it in interval notation:

$5x < 4x + 2$

Background

Topic: Linear Inequalities

This question tests your understanding of how to solve linear inequalities, use properties of inequalities, and express the solution both graphically and in interval notation.

Key Terms and Formulas:

• Additive Property of Inequality: If $a < b$, then $a + c < b + c$ for any real number $c$.

• Interval Notation: A way to represent the set of solutions using parentheses or brackets.

• Graphing Inequalities: Use a number line, shading the region that represents the solution set.

Step-by-Step Guidance

1. Start by identifying the inequality: $5x < 4x + 2$.

2. Apply the subtraction property of inequality: subtract $4x$ from both sides to isolate $x$.

   $5x - 4x < 4x + 2 - 4x$

3. Simplify both sides to get the variable by itself.

   $x < 2$

4. Think about how to represent this solution on a number line. Since $x$ is less than $2$(not equal), use a parenthesis at $2$ and shade to the left.

Try solving on your own before revealing the answer!

Final Answer: The solution set is $(-\infty, 2)$

In interval notation, this means all real numbers less than $2$, not including $2$ itself. The graph uses a parenthesis at $2$ and shades to the left.