Solving Linear Inequalities and Expressing Solutions in Interval Notation

Q1. Solve the following inequality and write the solution set using interval notation:

$5(3x + 10) > 5$

Background

Topic: Linear Inequalities

This question tests your understanding of how to solve linear inequalities, including using properties of inequalities and expressing the solution in interval notation.

Key Terms and Formulas

• Distributive Property: $a(b + c) = ab + ac$

• Multiplication Property of Inequality: If $a < b$ and $c > 0$, then $ac < bc$. If $c < 0$, then $ac > bc$.

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

• Interval Notation: Used to express the set of solutions to an inequality. Parentheses $( )$ indicate endpoints are not included; brackets $[ ]$ indicate endpoints are included.

Step-by-Step Guidance

1. Notice that both sides of the inequality have a common factor of 5. Use the multiplication property of inequality to divide both sides by 5, simplifying the inequality.

2. After dividing, you should have an inequality in the form $3x + 10 > 1$.

3. Use the addition property of inequality to isolate the variable term. Subtract 10 from both sides to get $3x > -9$.

4. To solve for $x$, divide both sides by 3 (since 3 is positive, the direction of the inequality does not change). This will give you an inequality in terms of $x$.

Try solving on your own before revealing the answer!

Final Answer: $(-3, \infty)$

After dividing by 3, you get $x > -3$. In interval notation, this is written as $(-3, \infty)$, which means all real numbers greater than $-3$.