Problem 58 - Algebra for College Students

Question

Solve each inequality and express the solution set using interval notation. \(-4(2 x-1)-3(x+2) \geq 0\)

Step-by-Step Solution

Answer

The solution set is \[(-\infty, -\frac{2}{11}]\].

Step 1: Distribute Multiplication

Start by distributing the multiplication within each term: Firstly, distribute \(-4\times(2x-1)\) to get: \(-4 \times 2x = -8x\) and \(-4 \times -1 = 4\), so it becomes \(-8x + 4\).Secondly, distribute \(-3\times(x+2)\) to get: \(-3 \times x = -3x\) and \(-3 \times 2 = -6\), so it becomes \(-3x - 6\).Combine these parts into the inequality: \[-8x + 4 - 3x - 6 \geq 0\].

Step 2: Combine Like Terms

Combine the like terms on the left side of the inequality:Combine \(-8x\) and \(-3x\) to get \(-11x\).Then combine the constant terms \(4\) and \(-6\) to get \(-2\).The inequality now becomes \[-11x - 2 \geq 0\].

Step 3: Isolate the Variable Term

Add 2 to both sides of the inequality to isolate the \(-11x\) term:\[-11x - 2 + 2 \geq 0 + 2\] simplifies to \[-11x \geq 2\].

Step 4: Solve for the Variable

Divide both sides of the inequality by \(-11\) to solve for \(x\). Remember, dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign.\[-11x \geq 2\] becomes \[x \leq -\frac{2}{11}\].

Step 5: Express in Interval Notation

Convert the solution \(x \leq -\frac{2}{11}\) into interval notation:Since \(x\) includes all values less than or equal to \(-\frac{2}{11}\), the interval notation is \[(-\infty, -\frac{2}{11}]\].

Key Concepts

Interval Notation

Interval notation is a handy way to express sets of numbers, particularly solutions to inequalities. It provides a simple representation of all the numbers that are included in an inequality.

• The **closed bracket** "]" or "]" indicates that an endpoint is included in the set (e.g., "\([a, b]\)"). In our case, \(-\frac{2}{11}\) is included in the set, represented by \((-\infty, -\frac{2}{11}]\).
• The **open bracket** "(") or ")" signifies that an endpoint is not included in the interval (e.g., "(a, b)"). For instance, infinity symbols \(\infty\) and \(-\infty\) always use open brackets "(") because they are not actual numbers and cannot be "reached".

Intervals allow for a concise expression of ranges that a variable like \(x\) can take, which, in this exercise, is all values less than or equal to \(-\frac{2}{11}\).

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a particular value or set of values. In understanding inequalities, representing the relationships between algebraic expressions is key.In the inequality given in the problem, \[-4(2x-1)-3(x+2) \geq 0\]Each part of the expression can be simplified or manipulated using algebraic rules.- **Variables** like \(x\), represent unknown numbers.- **Constants** like \(-4\) and \(-3\), stand alone and never change in value.- **Operations** like addition and multiplication, describe how values are combined.Understanding how to rearrange and combine these elements efficiently helps in solving such inequalities.

Distributive Property

The distributive property is a fundamental algebraic principle that allows you to multiply a single term by each term inside a bracket. You apply this property when an expression needs expansion.Given: \(-4(2x-1) \text{ and } -3(x+2)\)You'll use the distributive property to expand each expression:- **Left Term**: Apply \(-4\) to \(2x\) and \(-1\), giving \(-4 \times 2x = -8x\) and \(-4 \times -1 = 4\). Thus, the expression becomes \(-8x + 4\).- **Right Term**: Similarly, apply \(-3\) to \(x\) and \(2\), giving \(-3 \times x = -3x\) and \(-3 \times 2 = -6\). So, it becomes \(-3x - 6\).
Combining the results, the expression \[-8x + 4 - 3x - 6\]is more manageable and sets the stage for further simplification.

Solving Inequalities

Solving inequalities involves finding which values of a variable make the inequality true. It requires several steps including simplification, rearranging terms, and potentially reversing inequality signs when working with negative coefficients.Steps in the original solution:- **Combine like terms**: Identify and combine similar terms to simplify the inequality. Combining \(-8x\) with \(-3x\) yields \(-11x\), while constants \(4\) and \(-6\) yield \(-2\).- **Isolate the variable**: To solve \(-11x - 2 \geq 0\), add 2 to both sides to get \(-11x \geq 2\).- **Solve for the variable**: Divide each side by \(-11\) to isolate \(x\). Remember, division by a negative number reverses the inequality sign, changing it to \(x \leq -\frac{2}{11}\).This process illustrates how careful manipulation and a solid understanding of algebraic principles can solve complex inequalities, resulting in an accurately expressed interval notation.