Reasoning through inequality expressions | Linear inequalities | Algebra I | Khan Academy

TL;DR

Given positive integer a, negative integer b, and a/b > ab, the key conclusion is that b must be less than -1. Both expressions are negative, but a/b is less negative and has a smaller absolute value than ab. Multiplying the inequality by negative b reverses its direction, leading to 1 < b² and |b| > 1. Read on to see why each inequality step works.

Transcript

You go for a job interview. And the first thing that your interviewer says is, look, you have great work experience. You seem like a nice young person. But what I really care about is your logical reasoning capabilities. So what she says is, just sit down. I'm going to ask you a question about some math expressions. And you say, sure, shoot away. G... Read More

Key Insights

• 😑 The relationship between positive and negative integers in math expressions can provide valuable clues about the values of variables.
• 😑 Manipulating inequalities algebraically allows for a deeper understanding of the relationships between different mathematical expressions.
• 💁 Constraints and given information can help narrow down the possible values of variables in math problems.

Questions & Answers

Q: What can be concluded from a > 0, b < 0, and a/b > ab?

The conditions imply that b must be less than -1. They also show that a/b and ab are both negative, with a/b being the greater, less negative value.

Q: Why are a/b and ab both negative?

The integer a is positive, while b is negative. Dividing or multiplying a positive quantity by a negative quantity produces a negative result, so both a/b and ab are negative.

Q: What does a/b > ab mean on a number line?

Both quantities lie to the left of 0 because they are negative. Since a/b is greater, it lies to the right of ab and is therefore less negative.

Q: How do the absolute values of a/b and ab compare?

The absolute value of a/b is less than the absolute value of ab. On the number line, a/b is closer to 0, so its distance from 0 is smaller.

Q: Why does multiplying a/b > ab by b reverse the inequality?

The inequality reverses because b is less than 0. After multiplication, the result is a < ab².

Q: How does a < ab² simplify to 1 < b²?

Both sides can be divided by a because a is greater than 0. Dividing by a positive quantity does not reverse the inequality, leaving 1 < b².

Q: What does 1 < b² reveal about b?

It means that the absolute value of b is greater than 1. Equivalently, b must be less than -1 or greater than 1.

Q: Why must b be less than -1 rather than greater than 1?

The original conditions state that b is less than 0, so the possibility b > 1 is excluded. Combining b < 0 with |b| > 1 leaves only b < -1.

Summary & Key Takeaways

• In a job interview, a candidate is asked to analyze the relationship between two integers, a and b, based on the inequality a/b > a times b.

• By considering the signs of a and b, it is deduced that a/b is less negative than a times b.

• Manipulating the inequality further reveals that the absolute value of b must be greater than 1 and b must be less than 0.