Question

In Exercises 37–44, solve the inequality. $$ -2 \sqrt[3]{x+4}<12 $$

Short answer

The solution to the inequality is \(x > -220\)

Step-by-step solution

Isolate the Cube Root

Instead of dividing by -2, multiply both sides by -1/2. This changes the direction of the inequality. Therefore, \(\sqrt[3]{x+4}>-6\)

Eliminate the Cube Root

Cube both sides of the inequality to eliminate the cube root. This gives: \(x+4 > -216\)

Solve for x

Solve for x by subtracting 4 from both sides of the equation, which gives: \(x > -220\)

Key concepts

Cube Root

Understanding the concept of the cube root is essential in algebra, particularly when solving algebraic inequalities. The cube root of a number \(x\) is a value that, when multiplied by itself three times, equals \(x\). Unlike square roots, cube roots can yield both positive and negative results since multiplying three negative numbers results in a negative product. This nature of cube roots is vital when dealing with inequalities because it can affect the direction of the inequality when manipulated. To calculate the cube root of \(x\), we write it as \(\sqrt[3]{x}\), which is an operation asking "what number, when cubed, gives \(x\)?" For instance, \(\sqrt[3]{8} = 2\) because \(2 \times 2 \times 2 = 8\). Recognizing this operation allows us to handle cube-root expressions in algebraic equations and inequalities confidently.

Solving Algebraic Inequalities

When solving algebraic inequalities, a series of logical steps can help reach the solution. The goal is to isolate the variable on one side of the inequality. This process involves similar operations as when solving equations, but with one crucial difference: reversing the inequality symbol when multiplying or dividing by a negative number.
Here is a structured approach to solve an inequality like \(-2 \sqrt[3]{x+4}<12\):

• Isolate the term: Start by manipulating the inequality to isolate terms involving the variable. Here, multiply each side by \(-1/2\), which reverses the inequality to \(\sqrt[3]{x+4}>-6\).
• Remove the cube root: Cube both sides to eliminate the cube root. This gives \((x+4) > (-6)^3\), which simplifies to \(x+4 > -216\).
• Simplify to find the variable: Solve for \(x\) by subtracting 4 from both sides resulting in \(x > -220\).

The process demonstrates the logical sequence required. Remember to carefully handle operations that might affect the inequality, especially when working with negative numbers.

Algebra 2

Algebra 2 extends the concepts learned in Algebra 1, introducing more complex functions and topics such as inequalities, cube roots, and various types of polynomials. The course builds on students' understanding of algebraic principles and pushes towards a deeper comprehension of mathematical structures.
In the context of Algebra 2, it’s not just about finding a solution, but also interpreting what the solution signifies in a real-world context. For example, solving the inequality \(x > -220\) involves understanding that the solution represents all numbers greater than \(-220\). This broad range of possibilities is often what makes Algebra 2 exciting yet challenging. It requires logical reasoning, flexibility in problem-solving, and often incorporates various methods and strategies to find solutions. By exploring these topics, students develop a robust foundation for calculus and other advanced mathematical studies.