Problem 60 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The solution set is \\( (-\frac{7}{4}, \infty) \\).

Step 1: Distribute terms

Start by applying the distributive property to each side of the inequality. For the left side, distribute 3 across the expression \(x-1\), and for the right side, distribute -2 across \(x+4\). This will transform the equation into: \[3x - 3 - x + 2 > -2x - 8.\]

Step 2: Simplify both sides

Combine like terms on both sides of the inequality. On the left, \(3x\) and \(-x\) combine to \(2x\), and \(-3\) and \(+2\) combine to \(-1\). The inequality now becomes: \[2x - 1 > -2x - 8.\]

Step 3: Isolate variable term on one side

Add \(2x\) to both sides to move all the x terms to one side of the inequality: \[2x + 2x - 1 > -8.\] Simplifying gives: \[4x - 1 > -8.\]

Step 4: Isolate constant

Add 1 to both sides to solve for the term without x: \[4x - 1 + 1 > -8 + 1.\] This simplifies to \[4x > -7.\]

Step 5: Solve for x

Divide both sides by 4 to solve for x: \[x > -\frac{7}{4}.\]

Step 6: Express the solution in interval notation

Since \(x > -\frac{7}{4}\), the solution in interval notation is \((-\frac{7}{4}, \, \infty)\).

Key Concepts

Distributive Property

When you encounter an inequality like this one, you will often need to use the distributive property. This rule enables us to multiply a single term by each term within a set of parentheses. Let's have a closer look at how this works.
For the inequality \(3(x-1)-(x-2)>-2(x+4)\), we start by distributing each term:

• Distribute 3 to both \(x\) and \(-1\) in \(3(x-1)\), resulting in \(3x - 3\).
• Distribute -1 to both \(x\) and \(-2\) in \(-(x-2)\), yielding \(-x + 2\).
• Distribute -2 to both \(x\) and 4 in \(-2(x+4)\), giving \(-2x - 8\).

After applying the distributive property to each part, a complex expression becomes simpler and ready for the next operations.
This property is crucial because it maintains the balance in equations and inequalities, allowing clear steps toward the solution.

Interval Notation

Interval notation is a way of writing the set of all numbers that satisfy an inequality. It captures the span of possible solutions compactly. When solving inequalities, such as this one where the solution was found to be \(x > -\frac{7}{4}\), you express this in interval notation.
This notation uses parenthesis \(()\) or brackets \([]\) to describe the interval:

• Parentheses \(()\) are used to denote "less than" or "greater than," meaning the endpoint is not included.
• Brackets \([]\) represent "less than or equal to" and "greater than or equal to," which means the endpoint is included.
• The expression \((a, b)\) includes all numbers between \(a\) and \(b\), exclusive of \(a\) and \(b\) themselves.

For your solution \(x > -\frac{7}{4}\), it translates to \(( -\frac{7}{4}, \infty )\), encompassing all numbers greater than \(-\frac{7}{4}\) but not including \(-\frac{7}{4}\) itself.

Combining Like Terms

Combining like terms is a handy tool for simplifying algebraic expressions, which is especially useful in solving equations and inequalities.
In the inequality process, after distributing the terms, you need to look for terms with the same variable or constant that can be grouped together.
For the given problem, look at each side of the inequality and find similar terms to simplify the expression:

• On the left side \(3x - 3 - x + 2\), you combine \(3x\) and \(-x\) to get \(2x\) and \(-3\) and \(+2\) to get \(-1\).
• This simplification gives \(2x - 1\).

With like terms combined, your expressions are less cluttered, making it easier to pinpoint the next steps to solving your inequality.

Isolation of Variables

Isolating a variable means getting that variable by itself on one side of the equation or inequality. This is often a goal in algebra to find what values can satisfy an equation or inequality. To achieve this, you perform operations to simplify and rearrange expressions.
In the inequality \(2x - 1 > -2x - 8\):

• Add \(2x\) to both sides to gather all \(x\) terms together.
• This yields \(4x - 1 > -8\).
• Next, isolate \(x\) by adding 1 to both sides, giving \(4x > -7\).
• Finally, divide both sides by 4 to get \(x \) by itself: \(x > -\frac{7}{4}\).

By carefully isolating the variable, you create clear steps toward discovering what \(x\) must be greater than to satisfy the inequality, leading to your final solution.