Absolute value inequalities | Linear equations | Algebra I | Khan Academy

Summary

This video tutorial provides a clear, step-by-step guide to solving absolute value inequalities, a topic often seen as confusing in algebra. Using intuitive explanations and number line visualizations, it demonstrates how to interpret absolute values as distances from zero and translate inequalities into compound forms. The instructor solves progressively complex examples, from simple one-step problems to linear expressions with coefficients and fractions, highlighting both less-than and greater-than scenarios. Viewers are guided to understand the underlying logic, avoid memorization pitfalls, and express solutions in interval notation, ensuring they can confidently tackle a wide range of absolute value inequalities.

Takeaways

• 😀 Absolute value represents the distance from 0 on a number line.
• 😀 For inequalities like |x| < a, the solution set includes all numbers between -a and a.
• 😀 For inequalities like |x| > a, the solution set includes numbers less than -a or greater than a.
• 😀 Visualizing absolute value inequalities on a number line helps prevent confusion and reduces the need to memorize rules.
• 😀 Compound inequalities arise naturally when solving absolute value problems, and they can be solved step by step.
• 😀 Solving |f(x)| < a involves setting up the inequality -a < f(x) < a.
• 😀 Solving |f(x)| > a involves setting up two inequalities: f(x) < -a or f(x) > a.
• 😀 When solving absolute value inequalities with linear expressions, isolate the variable after setting up the inequality.
• 😀 Multiplying or dividing by positive numbers in inequalities does not change the direction of the inequality, but careful arithmetic is crucial to avoid mistakes.
• 😀 Interval notation and compound inequalities are interchangeable ways to represent solution sets of absolute value inequalities.
• 😀 Understanding the logic behind absolute value inequalities is more important than memorizing formulas, as it helps solve more complex problems intuitively.
• 😀 Visual examples and repeated practice help internalize the concepts of absolute value inequalities.

Q & A

What does theQ&A generation for script absolute value of a number represent?

The absolute value of a number represents its distance from 0 on the number line, regardless of direction.

How do you solve an absolute value inequality of the form |x| < a?

You translate it into a compound inequality: -a < x < a, because x must be less than a units away from 0.

How do you solve an absolute value inequality of the form |x| > a?

You split it into a disjoint inequality: x < -a or x > a, because x must be more than a units away from 0.

In the example |x| < 12, what is the solution set?

The solution set is all x values between -12 and 12, not including -12 and 12, written in interval notation as (-12, 12).

For the inequality |7x| ≥ 21, why do we consider two cases?

Because the absolute value measures distance from 0, |7x| ≥ 21 means 7x is either at least 21 units to the right (≥ 21) or 21 units to the left (≤ -21) of 0.

How do you solve the compound inequality -7 < 5x + 3 < 7?

Subtract 3Absolute value inequalities from all parts to get -10 < 5x < 4, then divide by 5 to find -2 < x < 4/5.

What is the solution set for |5x + 3| < 7 in interval notation?

The solution set is all x values between -2 and 4/5, written as (-2, 4/5).

How do you handle absolute value inequalities with fractions, such as |(2x/7) + 9| > 5/7?

Multiply through by the denominator to eliminate fractions, then split into two inequalities: one greater than 5/7 and one less than -5/7, and solve each for x.

Why is visualizing absolute value inequalities on a number line recommended?

Because it helps you intuitively understand the range or distance from 0, preventing confusion and reducing the need to memorize rules.

What general rule can be used for inequalities of the form |f(x)| < a or |f(x)| > a?

For |f(x)| < a → -a < f(x) < a; for |f(x)| > a → f(x) < -a or f(x) > a. These come from the concept of distance from 0.

How should you approach absolute value inequalities to avoid mistakes?

Focus on understanding the meaning of absolute value as distance, visualize on a number line, carefully isolate the variable, and check your arithmetic, especially with fractions or negatives.

What is the solution set for |(2x/7) + 9| > 5/7?

After solving, the solution set is x < -34 or x > -29, which in interval notation is (-∞, -34) ∪ (-29, ∞).