How to Solve Compound Inequalities in Algebra Easily

TL;DR

To solve compound inequalities, break them into separate parts and find the solution for each. For example, from the inequalities z < 4 and z ≥ -6, the solution set includes any value of z less than 4 or greater than or equal to -6. Always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Transcript

Solve for z. 5z plus 7 is less than 27 or negative 3z is less than or equal to 18. So this is a compound inequality. We have two conditions here. So z can satisfy this or z can satisfy this over here. So let's just solve each of these inequalities. And just know that z can satisfy either of them. So let's just look at this. So if we look at just th... Read More

Key Insights

• 🤪 A compound inequality consists of two conditions that z can satisfy, allowing for multiple possible values.
• ➗ When dividing or multiplying both sides of an inequality by a negative number, the inequality sign must be swapped.
• 🤪 The solution set of a compound inequality is determined by finding the values of z that satisfy either one or both 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.