Answered

Amy solved the exponential inequality

$4^{2 x-2} \geq 2$

by first rewriting the inequality as

$4^{2 x-2}-2 \geq y$

and then graphing this inequality. She determines the solution set of the inequality is $(-\infty, 1.25]$.



Which best describes Amy's solution?


AAmy is correct. The interval $(-\infty, 1.25]$ is the solution set for the inequality.
BAmy did not break the inequality into two related inequalities. Her solution set only relates to one inequality.
CAmy used the incorrect inequality symbol. The solution set should be $[1.25, \infty)$.
DAmy used the wrong variable when writing the solution. The solution set should relate to $y$ and not $x$.

Verified Answer :

  • Solve the inequality \(4^{2x-2} \geq 2\) by expressing both sides as powers of 2: \(2^{4x-4} \geq 2^1\).
  • Simplify the inequality to \(4x-4 \geq 1\), which results in \(x \geq 1.25\).
  • Identify that the correct solution set is \([1.25, \infty)\).
  • Conclude that Amy used the incorrect inequality symbol, as her result \((-\infty, 1.25]\) is the inverse of the correct set. The best description is: \(\boxed{\text{Amy used the incorrect inequality symbol. The solution set should be } [1.25, \infty).}\)

Explanation

  1. Solving the Exponential Inequality

The goal is to solve the exponential inequality \(4^{2x-2} \geq 2\). We begin by expressing both sides of the inequality with the same base, which is 2. Note that \(4 = 2^2\), so we can substitute this into the expression:
\[(2^2)^{2x-2} \geq 2^1\]
Using the laws of exponents, \((a^m)^n = a^{m \cdot n}\), we simplify the left side:
\[2^{2(2x-2)} \geq 2^1\]
\[2^{4x-4} \geq 2^1\]
Since the base 2 is greater than 1, the inequality holds for the exponents:
\[4x - 4 \geq 1\]
Adding 4 to both sides gives:
\[4x \geq 5\]
Dividing by 4 gives:
\[x \geq 1.25\]
Thus, the correct solution set for the inequality \(4^{2x-2} \geq 2\) is the interval \([1.25, \infty)\).

  1. Analyzing Amy's Error

Amy determined the solution set to be \((-\infty, 1.25]\). Comparing this result to our analytical solution of \([1.25, \infty)\), it is clear that Amy's result is incorrect. The interval she found represents the opposite range of values compared to the correct solution. This discrepancy indicates that the inequality symbol used in her analysis or graphing process was likely reversed, leading to the set of values that satisfy the condition \(\leq\) rather than \(\geq\). Therefore, the statement "Amy used the incorrect inequality symbol. The solution set should be [1.25, \infty)" correctly describes her solution.

Examples

Exponential inequalities appear in scenarios such as calculating the time required for an investment to grow beyond a certain target amount. For instance, if an account earns interest, you might need to determine when the balance exceeds a specific value. Solving these correctly ensures accurate financial planning and projections. Understanding the correct direction of an inequality sign is crucial, as it determines whether you are looking for the period before or after reaching a certain milestone.

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