Compound inequalities  Linear inequalities  Algebra I  Khan Academy  Summary and Q&A
TL;DR
Learn how to solve compound inequalities by finding the possible values of z that satisfy the given conditions.
Questions & Answers
Q: How do you solve the compound inequality 5z + 7 < 27 or 3z ≤ 18?
To solve this compound inequality, we solve each inequality separately. For 5z + 7 < 27, we isolate z and divide both sides by 5. For 3z ≤ 18, we divide both sides by 3 and swap the inequality sign. The solution set is z < 4 or z ≥ 6.
Q: Why do we swap the inequality sign when dividing by a negative number?
When dividing or multiplying both sides of an inequality by a negative number, we swap the inequality sign to maintain the correct order of the inequality. Swapping the sign ensures the solution set remains unchanged.
Q: Can you explain the number line representation of the solution set?
The number line is used to represent the values of z that satisfy the compound inequality. For z < 4, a circle is placed around 4 (excluding it), and all values to the left of 4 are included. For z ≥ 6, a circle is placed around 6 (including it), and all values to the right of 6 are included. The shaded region on the number line represents the solution set.
Q: How can we determine if a specific number satisfies the compound inequality?
To determine if a number satisfies the compound inequality, substitute it into both inequality conditions. If it satisfies either one or both conditions, then it is a valid solution. If it fails to satisfy both conditions, it is not a valid solution. Testing multiple numbers can help confirm the validity of the solution set.
Summary & Key Takeaways

A compound inequality consists of two conditions that z can satisfy, where z can satisfy either one or both of them.

The first inequality is solved by isolating z and dividing both sides by 5, resulting in z < 4.

The second inequality requires dividing both sides by 3 and swapping the inequality sign, resulting in z ≥ 6.

The solution set for the compound inequality is z < 4 or z ≥ 6, meaning any value of z less than 4 or greater than or equal to 6 satisfies the conditions.