Problem 12
Question
For the following exercises, solve the inequality. Write your final answer in interval notation. $$ -3(2 x+1)>-2(x+4) $$
Step-by-Step Solution
Verified Answer
The solution in interval notation is
\((-\infty, \frac{5}{4})\).
1Step 1: Distribute the Terms
First, distribute the terms on both sides of the inequality. On the left side, distribute the \(-3\) into \((2x + 1)\), resulting in \(-6x - 3\). On the right side, distribute the \(-2\) into \((x + 4)\), resulting in \(-2x - 8\). The inequality now becomes: \[-6x - 3 > -2x - 8\]
2Step 2: Move All Terms Involving x to One Side
Add \(6x\) to both sides of the inequality to move all terms involving \(x\) to one side: \[-6x + 6x - 3 > -2x + 6x - 8\]This simplifies to:\[-3 > 4x - 8\]
3Step 3: Move Constant Terms to the Other Side
Add \(8\) to both sides to move constant terms to the other side of the inequality:\[-3 + 8 > 4x - 8 + 8\]This simplifies to:\[5 > 4x\]
4Step 4: Solve for x
To isolate \(x\), divide both sides of the inequality by \(4\):\[\frac{5}{4} > x\]Or equivalently, \[x < \frac{5}{4}\]
5Step 5: Write the Solution in Interval Notation
The solution, \(x < \frac{5}{4}\), can be expressed in interval notation as:\((-\infty, \frac{5}{4})\). This indicates that \(x\) includes all values less than \(\frac{5}{4}\).
Key Concepts
Interval NotationDistributive PropertyAlgebraic Manipulation
Interval Notation
Interval notation is a way of representing a set of numbers between specified boundaries, perfect for expressing solutions to inequalities. It's like giving directions to where a number can "live." There are two key symbols used in interval notation: parentheses and brackets.
When solving inequalities and expressing answers in interval notation, pay close attention to whether the inequality sign is strict (< or >), meaning parentheses are used, or inclusive (≤ or ≥), indicating brackets.
In the exercise solution given, the inequality is expressed as \(x < \frac{5}{4}\), which translates to the interval (-∞, \(\frac{5}{4}\)) in interval notation. Here, the parenthesis before \(\frac{5}{4}\) signifies that \(\frac{5}{4}\) is not included in the solution set.
- Parentheses ( ) indicate that the number immediately next to it is not included.
- Brackets [ ] indicate that the number immediately next to it is included in the set.
When solving inequalities and expressing answers in interval notation, pay close attention to whether the inequality sign is strict (< or >), meaning parentheses are used, or inclusive (≤ or ≥), indicating brackets.
In the exercise solution given, the inequality is expressed as \(x < \frac{5}{4}\), which translates to the interval (-∞, \(\frac{5}{4}\)) in interval notation. Here, the parenthesis before \(\frac{5}{4}\) signifies that \(\frac{5}{4}\) is not included in the solution set.
Distributive Property
The distributive property is a vital algebraic principle used to simplify expressions, especially when you encounter parentheses in inequalities or equations. Understanding this concept allows you to break down expressions into manageable parts.
Simply put, the distributive property means you can multiply a number by each term inside parentheses. For instance, if you have the expression \(a(b + c)\), it can be expanded using the distributive property to \(ab + ac\).
In the exercise, both sides of the inequality required the use of distributive property:
Simply put, the distributive property means you can multiply a number by each term inside parentheses. For instance, if you have the expression \(a(b + c)\), it can be expanded using the distributive property to \(ab + ac\).
In the exercise, both sides of the inequality required the use of distributive property:
- On the left side: \(-3(2x + 1)\), which becomes\(-6x - 3\) after applying the distributive property.
- On the right side: \(-2(x + 4)\), which simplifies to\(-2x - 8\).
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying algebraic expressions or inequalities to solve for variables. This process often includes moving terms involving variables to one side, and constant terms to the opposite side of an equation or inequality.
Consider the exercise solution where we aim to isolate the variable \(x\). This helps to better understand which numbers satisfy the inequality.
Consider the exercise solution where we aim to isolate the variable \(x\). This helps to better understand which numbers satisfy the inequality.
- First, terms involving \(x\) are gathered on one side by adding or subtracting them across the inequality. In the solution, 6x was added to both sides, resulting in the inequality: -3 > 4x - 8.
- Next, constant terms are moved to the opposite side by adding or subtracting them across the inequality. In this example, 8 is added to both sides, simplifying the inequality to 5 > 4x.
- Finally, to solve for x, both sides of the inequality are divided by 4 to find the value of x: \(x < \frac{5}{4}\).
Other exercises in this chapter
Problem 11
For the following exercises, solve the equation for \(x\). $$ \frac{x}{3}-\frac{3}{4}=\frac{2 x+3}{12} $$
View solution Problem 11
For each of the following exercises, solve the equation for y in terms of \(x\). $$ 4 x+2 y=8 $$
View solution Problem 12
For the following exercises, solve the rational exponent equation. Use factoring where necessary. $$ x^{\frac{7}{3}}-3 x^{\frac{4}{3}}-4 x^{\frac{1}{3}}=0 $$
View solution Problem 12
For the following exercises, solve the quadratic equation by factoring. $$ 8 x^{2}+6 x-9=0 $$
View solution