Problem 17 - Algebra 2

Question

Solve each inequality. Check your solution. $$ 4^{4 a+6} \leq 16^{a} $$

Step-by-Step Solution

Answer

The solution is \(a \leq -3\).

Step 1: Rewrite the Equation with Same Base

First, note that both 4 and 16 are powers of 2: \( 4 = 2^2 \) and \( 16 = 2^4 \). Rewrite the inequality using these bases:\[(2^2)^{4a+6} \leq (2^4)^{a}\]

Step 2: Simplify Exponents

Apply the power of a power rule, which states \((a^m)^n = a^{m \cdot n}\). Therefore, the inequality becomes:\[2^{2(4a+6)} \leq 2^{4a}\]Simplify further:\[2^{8a + 12} \leq 2^{4a}\]

Step 3: Solve the Exponent Inequality

Since the bases are the same, compare the exponents:\[8a + 12 \leq 4a\]Subtract \(4a\) from both sides:\[ 4a + 12 \leq 0\]

Step 4: Isolate the Variable

Subtract 12 from both sides to isolate the terms involving \(a\):\[4a \leq -12\]Then divide by 4:\[a \leq -3\]

Step 5: Check the Solution

To verify, choose any number for \(a\) less than or equal to -3, such as -3 itself. Substitute \(a = -3\) back into the original inequality:\[4^{4(-3)+6} \leq 16^{-3}\]Simplify:\[4^{-6} \leq 16^{-3}\]Convert to base 2:\[(2^2)^{-6} = 2^{-12} \quad \text{and} \quad (2^4)^{-3} = 2^{-12}\]Since \(2^{-12} = 2^{-12}\), the solution satisfies the inequality.

Key Concepts

Solving Inequalities

To solve inequalities, you first need to understand what makes them different from equations. Inequalities demonstrate a range rather than a specific value.
They use symbols like `≤` (less than or equal to), `≥` (greater than or equal to), `<` (less than), and `>` (greater than). For example, if we have an inequality \(4x + 5 \leq 9\), this means that there are multiple solutions for \(x\) which make the statement true.
The method for solving inequalities is similar to solving regular equations. Still, there are some rules to be aware of, such as flipping the inequality sign when multiplying or dividing both sides by a negative number.
In our specific problem, the inequality involves exponents, but the process starts by transforming it into an equation first by having equal bases. This makes it easier to handle since comparing the exponent parts provides the solution.

Exponents

Exponents are an essential part of algebra that provide a way to express repeated multiplication. When understanding exponents, remember these rules:

• Product of Powers Rule: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power Rule: \((a^m)^n = a^{m\cdot n}\)
• Power of a Product Rule: \((ab)^n = a^n b^n\)

In the exercise, the `Power of a Power Rule` was used. Both numbers 4 and 16 were rewritten in terms of one base, 2. This allows for straightforward application of the exponential laws to simplify the problem and focus solely on the exponents. Simplifying exponential expressions is crucial for solving complex inequalities efficiently.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations or inequalities to isolate variables.
The key steps in this process include adding, subtracting, multiplying, or dividing terms on both sides of an equation.
In our task, after rewriting with a common base and simplifying, we were left with the expression \(8a + 12 \leq 4a\).
This required isolating \(a\) by performing basic algebraic operations:

• Subtract \(4a\) from both sides to get: \(4a + 12 \leq 0\).
• Subtract 12 from both sides: \(4a \leq -12\).
• Divide by 4: \(a \leq -3\).

These steps simplify any complex equation, allowing for a solution to be found. Always verify your final answer by substituting back into the original problem, ensuring the inequality holds true, as illustrated in the problem where \(a = -3\) checked out correctly.