Problem 64 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ -2(4 x-1)<3(5 x+8) $$

Step-by-Step Solution

Answer

Answer: The solution to the inequality is $x>\frac{-22}{23}$.

Step 1: Distribute numbers outside the parentheses

Distribute -2 to both terms in the first parentheses and 3 to both terms in the second parentheses: $$ -8x+2<15x+24 $$

Step 2: Combine like terms

There are no like terms on each side of the inequality to combine at this point.

Step 3: Isolate x

Add 8x to both sides of the inequality to gather all the x terms on the right side and the constants on the left side: $$ 2<23x+24 $$ Now, subtract 24 from both sides of the inequality: $$ -22<23x $$

Step 4: Solve for x

Divide by 23 on both sides to solve for x: $$ \frac{-22}{23}\frac{-22}{23} $$

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with variables and numbers to find unknowns. It uses symbols, often letters, to represent numbers in equations and expressions. Understanding how to manipulate these symbols is key to solving problems in algebra. The goal is to simplify expressions and solve equations or inequalities by following specific rules.

• Variables are symbols like \(x\) or \(y\) that represent unknown numbers.
• Algebraic expressions combine variables and constants (known numbers) using operations like addition, subtraction, multiplication, and division.
• Equations are statements that two expressions are equal, while inequalities show a relationship where one expression is greater or less than the other.

Working with algebraic expressions and equations involves learning how to apply these basic operations and rules to solve for the unknowns.

Solve Inequalities

Solving inequalities in algebra is similar to solving equations but with a few key differences. Inequalities are mathematical statements that show the relation of one expression being greater or less than another.

Instead of finding an exact solution, like with equations, solving an inequality means finding a range of values that satisfy the inequality. For example, inequality solutions are often presented in a format such as \(x > a\) or \(x < b\). To solve inequalities:

• Apply the same operations to both sides of the inequality to maintain balance, just like solving equations.
• Be cautious when multiplying or dividing by a negative number—it reverses the inequality sign.
• Check the solution by plugging in values to ensure the inequality holds true.

Knowing how to interpret and solve inequalities broadens your ability to handle different mathematical challenges.

Distribution in Algebra

The distribution property is a crucial algebraic tool that simplifies expressions and equations by eliminating parentheses. When a number (coefficient) is outside a set of parentheses, you "distribute" it by multiplying it with each term inside the parentheses.
For example, the expression \(-2(4x - 1)\) should be distributed as follows:
- Multiply \(-2\) by \(4x\) to get \(-8x\).- Multiply \(-2\) by \(-1\) to get \(+2\).
This transforms the expression within the inequality into a simpler form: \(-8x + 2\). Distributing is essential when working with inequalities because it makes the expressions easier to manipulate, helping you to solve for the unknown variable more efficiently.

Isolate Variables

Isolating the variable is a strategy used to solve equations and inequalities by getting the variable alone on one side of the equation or inequality sign. This process involves strategic operations that move all terms involving the variable to one side and constants to the other.
Here’s how you can isolate variables step-by-step:

• Add or subtract terms on both sides to move constants and variables where you want them. For example, adding \(8x\) on both sides helps in collecting all \(x\)-terms.
• After that, adjust the inequality to have only the variable on one side. For instance, subtraction of \(24\) after adding \(8x\) aligns constants to one side.
• Finally, divide or multiply to solve for the variable, ensuring to reverse the inequality sign if dividing or multiplying by a negative. Dividing by \(23\) solves for \(x\) in the inequality.

Understanding these techniques is essential in becoming proficient in solving equations and inequalities, simplifying your path to find the correct solution.