Problem 50 - Algebra for College Students

Question

Solve each inequality and express the solution set using interval notation. \(9 x+5<6 x-10\)

Step-by-Step Solution

Answer

The solution set is \((-\infty, -5)\).

Step 1: Move Variables to One Side

Start by getting all the terms with \(x\) on one side of the inequality. Subtract \(6x\) from both sides to isolate the variable terms on one side.\[9x + 5 - 6x < 6x - 10 - 6x\]This simplifies to:\[3x + 5 < -10\]

Step 2: Isolate the Term with Variable

Now, get \(3x\) by itself by removing the constant term on the same side. Subtract 5 from both sides.\[3x + 5 - 5 < -10 - 5\]This simplifies to:\[3x < -15\]

Step 3: Solve for Variable

Divide each side of the inequality by 3 to solve for \(x\).\[\frac{3x}{3} < \frac{-15}{3}\]This simplifies to:\[x < -5\]

Step 4: Express in Interval Notation

The solution to the inequality \(x < -5\) can be expressed in interval notation as:\[(-\infty, -5)\]

Key Concepts

Interval Notation

Interval notation is a method to describe ranges of values that satisfy a particular condition, such as solutions to inequalities. When expressing solutions, interval notation provides a compact and efficient format. For instance, the interval \((-\infty, -5)\) indicates all real numbers less than \(-5\).

In interval notation:

• Parentheses \(( )\) are used to denote that an endpoint is not included in the set (open interval).
• Brackets \([ ]\) are used to show that an endpoint is included (closed interval).
• A comma separates the lower bound from the upper bound, such as \((a, b)\) for \(a < x < b\).
• If the interval extends infinitely in one direction, infinity \(\infty\) with a parenthesis is used, since infinity is not a specific number.

Understanding interval notation allows you to visualize and communicate solutions to inequalities succinctly.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations or inequalities to make them easier to solve. This involves a sequence of operations that maintain the truth of an inequality or equation.

Here is how it generally works:

• **Move Variables to One Side**: Combine like terms by getting all variable terms on one side and constant terms on the other. This can involve adding or subtracting terms from both sides of the inequality.
• **Isolate the Variable Term**: Further simplify by eliminating any constants alongside the variable. This could involve subtracting numbers from both sides.
• **Solve for the Variable**: Finally, get the variable by itself by dividing or multiplying both sides by the same number, remembering to reverse the inequality sign if multiplying or dividing by a negative.

Algebraic manipulation is essential for unraveling complex mathematical statements and transforming them into a solvable format.

Inequality Solutions

Solving inequalities requires understanding how to manipulate and interpret algebraic expressions to find all possible solutions that make the inequality true.

Here is a step-by-step approach:

• Start with simplifying each side of the inequality as needed, combining like terms and constants.
• Rearrange the inequality to isolate the variable on one side using algebraic manipulation.
• Pay special attention to the properties of inequalities:
  • If you multiply or divide both sides by a negative number, flip the inequality sign.
• Once the variable is isolated, check if additional steps like factoring or simplifying are needed.
• The final result can be written in interval notation, offering a clear view of the solution set.

Through careful manipulation and understanding, students can determine the range of values fulfilling the inequality conditions.