In Exercises 51–58, solve each compound inequality. 6 < x + 3 < 8

Verified step by step guidance

1. Understand that the compound inequality 6 < x + 3 < 8 means that x + 3 is greater than 6 and less than 8 at the same time.
2. To isolate x in the middle, subtract 3 from all three parts of the inequality: 6 - 3 < x + 3 - 3 < 8 - 3.
3. Simplify each part: 3 < x < 5.
4. Interpret the solution: x is any number greater than 3 and less than 5.
5. Express the solution in interval notation as (3, 5).

Key Concepts

Compound Inequalities

A compound inequality involves two inequalities joined together, often with 'and' or 'or'. In this case, the inequality 6 < x + 3 < 8 means x + 3 is greater than 6 and less than 8 simultaneously. Solving compound inequalities requires working on both parts at the same time.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side by performing inverse operations such as addition, subtraction, multiplication, or division. When solving 6 < x + 3 < 8, subtract 3 from all parts to maintain the inequality and find the range of x.

Properties of Inequalities

Properties of inequalities dictate how inequalities behave under operations. For example, adding or subtracting the same number from all parts of an inequality does not change its direction. Multiplying or dividing by a positive number also preserves the inequality, which is essential when manipulating compound inequalities.