Problem 43 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. $$ |x| \leq 3 $$

Step-by-Step Solution

Answer

The solution to the inequality is all real numbers, represented in interval notation as \((-\

Step 1: Distribute

First, distribute the 3 on the left-hand side of the inequality. \[3(2x - 4) - 4x < 2x + 3\]

Step 2: Simplify

Multiply out the 3 on the left-hand side: \[6x - 12 - 4x < 2x + 3\]

Step 3: Combine like terms

Combine the \(6x\) and \(-4x\) on the left-hand side: \[2x - 12 < 2x + 3\]

Step 4: Move terms to one side

Subtract \(2x\) from both sides: \[-12 < 3\]

Step 5: Solve and interpret

Since \(-12 < 3\) is always true, the inequality holds for all real numbers, hence we represent this in interval notation as \((-\infty, \infty) \).

Key Concepts

The Distributive Property

The distributive property is key in solving many algebra problems, including inequalities. It allows you to multiply a single term by each term inside parentheses. In our original exercise, we start with the inequality 3(2x - 4) - 4x < 2x + 3. Using the distributive property, the term 3 is multiplied by both 2x and -4: 3 * 2x = 6x and 3 * -4 = -12. So, after distribution, the inequality becomes: 6x - 12 - 4x < 2x + 3. Distributing correctly sets the stage for the next steps.

Combining Like Terms

Once the distributive property has been applied, the next step is to combine like terms. This means we add or subtract terms that have the same variable part. In the inequality: 6x - 12 - 4x < 2x + 3, we can combine the terms involving x on the left-hand side (6x and -4x): 6x - 4x = 2x. The simplified inequality now reads: 2x - 12 < 2x + 3. Combining like terms helps to reduce the complexity of the inequality, making it easier to solve.

Understanding Interval Notation

Once an inequality is solved, the solution is often expressed using interval notation. In our given exercise, the inequality simplifies to -12 < 3. This statement is always true, meaning the inequality holds for all real numbers. Hence, in interval notation, we express this as (-∞, ∞). Interval notation uses brackets and parentheses to describe all the numbers that satisfy the inequality. Parentheses, ( and ), indicate that the endpoints are not included. Brackets, [ and ], indicate that the endpoints are included.

Algebraic Inequalities

Solving algebraic inequalities involves finding the values of the variable that make the inequality true. For our problem, we started with 3(2x - 4) - 4x < 2x + 3 and after distributing, combining like terms, and simplifying, it became evident that the inequality simplifies to a true statement -12 < 3. This means the inequality is true for all real numbers. Therefore, the solution set is all real numbers, expressed as (-∞, ∞) in interval notation. Remember these steps to solve most algebraic inequalities:

• Use distributive property to remove parentheses.
• Combine like terms to simplify the expression.
• Move all variable terms to one side.
• Solve the inequality like an equation.
• Express the solution in appropriate notation.

Solving these inequalities correctly assures a strong foundation in algebra.