Problem 72 - Algebra Form and Function

Question

In Problems \(71-74,\) decide for what value(s) of the constan \(A\) (if any) the equation has (a) The solution \(x=1\) (b) A solution \(x>1\) (c) No solution $$ 4 x^{3}=A $$

Step-by-Step Solution

Answer

Answer: (a) If \(A = 4\), the equation has the solution \(x = 1\). (b) If \(A > 4\), the equation has a solution with \(x > 1\). (c) There is no value of \(A\) that results in the equation having no solution.

Step 1: Find the value of A when x = 1

We are given the equation \(4 x^3 = A\). We are asked to find the value of \(A\) when the solution is \(x=1\). Substitute \(x=1\) into the equation: $$ 4(1)^3 = A $$ Simplify the equation: $$ 4 = A $$ So when \(x = 1\), the value of \(A\) is 4.

Step 2: Determine the conditions for a solution x > 1

Following the given equation \(4 x^3 = A\), we need to find under what conditions there exists a solution \(x > 1\). We know that \(x^3\) will be positive for any positive value of \(x\). Since we are looking for a solution where \(x > 1\), we can deduce that \(x^3 > 1\) and therefore, \(4 x^3 > 4\). From this, we get: $$ 4x^3 > 4 $$ Then, we can deduce that \(A > 4\) in order for the equation to have a solution for \(x > 1\).

Step 3: Determine the conditions for no solution

Since the equation \(4x^3 = A\) will always have a solution for some value of \(A\), there is no value of \(A\) that will result in the equation having no solution. #Answer#: (a) If \(A = 4\), the equation has the solution \(x = 1\). (b) If \(A > 4\), the equation has a solution with \(x > 1\). (c) There is no value of \(A\) that results in the equation having no solution.

Key Concepts

Solutions of Equations

When dealing with equations, finding solutions means identifying the values for variables that make the equation true. In the case of our cubic equation, \(4x^3 = A\), we are hunting for the value of \(x\) that satisfies the equation for a given constant \(A\).
This involves directly substituting potential values of \(x\) into the equation and seeing if the left-hand side equals the right-hand side.

• For instance, if you substitute \(x = 1\), the equation becomes \(4(1)^3 = A\), simplifying to \(4 = A\).
• If \(x = 1\) is indeed a solution, then \(A\) must be 4.
• Thus, any potential solution needs thorough verification by substitution.

Finding solutions this way helps unravel the nature of the equation and it's essential to explore multiple values at times to ensure any hidden solutions don't slip by.

Values of Constants

The constant \(A\) in the equation \(4x^3 = A\) plays a critical role in determining both the existence and nature of solutions. By manipulating the equation with different values of \(A\), you can discover crucial properties of the cubic function and its solutions.
The value of \(A\) essentially dictates the balance point for equation validity:

• When \(A = 4\), the equation specifically allows the solution \(x = 1\).
• When \(A > 4\), the equation permits solutions where \(x > 1\), meaning the value of \(x\) has increased to satisfy greater \(A\).
• In general, understanding the value \(A\) helps predict whether solutions are possible or if further exploration is needed.

Observing changes in \(A\) is a powerful reminder of the influence constants exert on equations, shaping both result possibilities and solution characteristics.

Inequalities in Equations

The study of inequalities within equations helps us establish constraints and realize conditions under which solutions exist or do not exist. Inequality plays a pivotal part in deciding solutions for the equation \(4x^3 = A\), especially when investigating beyond solitary solutions.
Let's dig into how inequalities fit in:

• If \(A > 4\), the inequality \(4x^3 > 4\) arises, indicating that \(x\) must be greater than 1.
• Such inequalities guide us to find solutions \(x > 1\), offering insights on finding solution sets rather than isolated numbers.
• Even though no value of \(A\) makes the equation unsolvable, playing with inequalities aids in understanding solution ranges and possibilities.

Thus, coupling equations with inequality concepts opens a broader spectrum for analyzing solutions, extending beyond static equals to dynamic conditions of greater than or less than.