Problem 47 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The solution is \([\frac{5}{12}, \infty)\).

Step 1: Distribute the 4

First, distribute the 4 across the terms inside the parenthesis in the inequality. This results in: \[ 4(3x - 2) \geq -3 \] \[ 12x - 8 \geq -3 \].

Step 2: Add 8 to Both Sides

Next, add 8 to both sides of the inequality to isolate the term with \(x\):\[ 12x - 8 + 8 \geq -3 + 8 \] \[ 12x \geq 5 \].

Step 3: Solve for x

Now, divide both sides by 12 to solve for \(x\):\[ \frac{12x}{12} \geq \frac{5}{12} \] \[ x \geq \frac{5}{12} \].

Step 4: Express in Interval Notation

Finally, express the solution set using interval notation, which represents all values of \(x\) that satisfy the inequality \(x \geq \frac{5}{12}\). Thus, the solution in interval notation is:\[ \left[ \frac{5}{12}, \infty \right) \].

Key Concepts

Interval Notation

Interval notation is a way to express a set of numbers between a start and an endpoint, which can include or exclude the points themselves. It's particularly useful in representing solution sets for inequalities. In this notation:

• Square brackets, \([a, b]\) denote that both endpoints are included in the set, also known as a closed interval.
• Parentheses, \( (a, b) \) indicate that the endpoints are not included, known as an open interval.
• Combinations are also possible, such as \([a, b)\) where \(a\) is included but \(b\) is not.
• To represent numbers extending infinitely in one direction, use infinity symbols \(\( \infty \)\) or negative infinity \(\( -\infty \)\), always with parentheses as infinity is not a number that can be included.

For example, the solution \([\frac{5}{12}, \infty)\) indicates all numbers \(x\) starting from \(\frac{5}{12}\) and going to infinity, including \(\frac{5}{12}\) itself.

Distribution

Distribution in the context of solving inequalities involves applying the distributive property of multiplication over addition or subtraction. This means multiplying each term inside a set of parentheses by a number outside the parentheses. The distributive property can be written as:\[ a(b + c) = ab + ac \]In the original exercise, you have:

• \(4(3x - 2) \geq -3\)

Using distribution, multiply every term inside the parentheses by 4:

• \(4 \cdot 3x - 4 \cdot 2 = 12x - 8\)

This simplifies the inequality, making it easier to isolate terms. The ability to distribute correctly is fundamental in turning complex inequalities into simpler ones that can be solved through basic algebraic methods. When you distribute correctly, it sets the stage for further steps in solving the inequality.

Solve Inequalities

Solving inequalities is similar to solving regular equations but with additional rules to maintain the inequality relationship. After distributing terms, the steps typically involve isolating the variable on one side using inverse operations, such as addition and division. Here’s how it works:

• First, ensure that like terms are joined. For instance, after distribution, the inequality \(12x - 8 \geq -3\) shows us we need to get \(x\) by itself.
• Add or subtract terms from both sides. Here, adding 8 to both sides simplifies it to \(12x \geq 5\).
• Divide both sides by the coefficient of \(x\) to solve for \(x\). Dividing by 12 gives \(x \geq \frac{5}{12}\).
• Critical to solving inequalities is remembering that if you multiply or divide by a negative number, you must flip the inequality sign. However, this was not needed in our example.

Following these steps with care ensures you solve the inequality correctly and express the solution both algebraically and, if needed, in interval notation.