SOLVING EXPONENTIAL INEQUALITIES || GRADE 11 GENERAL MATHEMATICS Q1

Summary

This educational video script teaches how to solve exponential inequalities. It explains that when the base (b) is greater than 1, the exponential function is increasing, and the inequality direction remains the same. If 0 < b < 1, the function is decreasing, and the inequality direction reverses. The script provides examples, illustrating how to manipulate the inequality by adding, subtracting, multiplying, or dividing by constants, and changing the inequality direction when involving negative numbers. It concludes with a quiz to test understanding.

Takeaways

• 📚 Exponential inequalities involve expressions with exponential functions.
• 🔢 If the base 'b' of the exponential function is greater than 1, the function is increasing, meaning if x < y, then b^x < b^y.
• 📉 If the base 'b' is between 0 and 1, the function is decreasing, so if x < y, then b^x > b^y.
• ➕ Adding or subtracting the same real number from both sides of an inequality does not change the inequality's direction.
• ➗ Multiplying or dividing both sides of an inequality by a positive real number retains the inequality's direction.
• ⤺ Multiplying or dividing both sides by a negative real number reverses the inequality's direction.
• 🌰 An example given is 3^(x+1) > 81, which simplifies to x+1 > 4, leading to the solution x > 3.
• 💡 For 2^(4x+1) ≤ 512, since 2^9 = 512, the inequality simplifies to 4x+1 ≤ 9, resulting in x ≤ 2.
• 🔄 When dealing with exponential functions with the same base, equate the exponents to solve the inequality.
• 🔄 For inequalities involving exponential functions with different bases, simplify and compare the exponents to find the solution.
• 🎓 The solution sets for exponential inequalities are expressed in terms of intervals, such as (3, ∞) or (-∞, 2].

Q & A

What is an exponential inequality?

An exponential inequality is an inequality that involves an exponential expression, typically in the form of b^x where b is the base and x is the exponent.

How does the direction of an inequality change when the base of the exponential function is greater than one?

If the base b is greater than one, the exponential function y = b^x is increasing for all values of x. This means that b^x < b^y if and only if x < y, so the direction of the inequality symbol is retained.

What happens to the direction of an inequality when the base of the exponential function is between zero and one?

If the base b is between zero and one, the exponential function y = b^x is decreasing for all values of x. This means that b^x > b^y if x < y, so the direction of the inequality symbol is reversed.

What rule applies when adding or subtracting the same real number from both sides of an inequality?

When the same real number is added or subtracted from both sides of an inequality, the sense of the inequality does not change.

How does multiplying or dividing both sides of an inequality by a positive real number affect the inequality?

Multiplying or dividing both sides of an inequality by the same positive real number does not change the sense of the inequality.

What is the effect of multiplying or dividing both sides of an inequality by a negative real number?

Multiplying or dividing both sides of an inequality by the same negative real number reverses the sense of the inequality.

In the example where 3^{x+1} > 81, what is the solution for x?

The solution for x is x > 3. This is because 3^4 = 81 and since the base is greater than one, the inequality direction is retained, leading to x + 1 > 4 and thus x > 3.

What is the solution set for the inequality 2^{4x+1} <= 512?

The solution set for x is x <= 2. Since 512 = 2^9 and the base is greater than one, the inequality direction is retained, leading to 4x + 1 <= 9 and solving gives x <= 2.

How do you solve the inequality 3^x < 9^{x-2}?

The inequality simplifies to 3^x < 3^{2x-4}. Since the base is greater than one, the inequality direction is retained, leading to x < 2x - 4 and solving gives x > 4.

In the example with 1/10^{x+5} >= 1/100^{3x}, what is the solution for x?

The solution for x is x >= 1. The inequality simplifies to 10^{-x-5} >= 10^{-6x}, and since the base is less than one, the inequality direction is reversed, leading to -x - 5 <= -6x and solving gives x >= 1.

What is the significance of the phrase 'the sense of the inequality is changed'?

The phrase 'the sense of the inequality is changed' means that when both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality symbol is reversed.