Problem 46 - College Algebra

Question

Solve each linear inequality in Exercises 27-48 and graph the solution set on a number line. Express the solution set using interval notation. $$\frac{4 x-3}{6}+2 \geq \frac{2 x-1}{12}$$

Step-by-Step Solution

Answer

The solution of the inequality \(x \geq -\frac{19}{6}\). The solution set in interval notation is \([-19/6, \infty)\).

Step 1: Simplify the equation

First, the given inequality \(\frac{4 x-3}{6}+2 \geq \frac{2 x-1}{12}\) should be simplified. In order to do this, all terms should be multiplied by 12, the least common multiple of the denominators, to remove the fractions. As a result, we get \(8x - 6 + 24 \geq 2x - 1\). Now combine the like terms, yielding \(8x + 18 \geq 2x - 1\) after simplifying.

Step 2: Solve for x

Now, let's isolate x by subtracting 2x from both sides, resulting in \(6x + 18 \geq - 1\). Next, subtract 18 from both sides, to get \(6x \geq -19\). Finally, divide both sides by 6 to solve for x, ending up with \(x \geq -\frac{19}{6}\) as the solution for the inequality.

Step 3: Write the solution in interval notation and on a number line

The solution in interval notation will be \([-19/6, \infty)\). For the number line, draw a closed dot (indicating inclusion) on -19/6 and draw a line extending to the right, towards positive infinity.

Key Concepts

Interval Notation

Interval notation is a way of representing solution sets for inequalities. When we solve an inequality like \( x \geq -\frac{19}{6} \), we often want to express the range of values that are included in the solution.
To do this succinctly, we use interval notation. The solution \( x \geq -\frac{19}{6} \) includes all numbers greater than or equal to \(-\frac{19}{6}\).

• The square bracket \([ \) means "including" the endpoint. So \([-\frac{19}{6}, \infty) \) indicates that the range starts at \(-\frac{19}{6}\) and goes up to infinity, including \(-\frac{19}{6}\) itself.
• In contrast, a parenthesis \(( \) would mean the endpoint is not included. For example, if the symbol were a parenthesis, \((-\frac{19}{6}, \infty) \) would mean \(-\frac{19}{6}\) is not part of the solution set.

Interval notation is a helpful shorthand when dealing with linear inequalities, making it easy to specify when endpoints are part of the solution set or not.

Inequality Graphing

Graphing linear inequalities provides a visual representation of solutions on a number line.
Let's break down how you would do this:

• First, draw a number line. This line acts as your visual framework.
• Locate the key value from your inequality; in this case, \(-\frac{19}{6}\).
• Since our inequality is \( x \geq -\frac{19}{6} \), this means \(-\frac{19}{6}\) is included. Represent this with a filled (or closed) dot on the number line.
• With the dot marking \(-\frac{19}{6}\), draw a line extending to the right to show all the numbers greater than \(-\frac{19}{6}\). This indicates that every number in that direction satisfies the inequality.

Graphing inequalities effectively illustrates the range of solutions and aids in understanding whether specific values should be included. This visual approach complements algebraic solutions.

Solving Inequalities

Solving linear inequalities is similar to solving linear equations, but with additional rules when multiplying or dividing by negative numbers. In the problem \( \frac{4x-3}{6} + 2 \geq \frac{2x-1}{12} \), the aim is to isolate the variable \(x\).
Here’s a simplified guide to solving such inequalities:

• Remove fractions: Multiply all terms by the least common multiple of the denominators. This eliminates fractions and simplifies subsequent steps. In this case, multiplying by 12 removes the denominators.
• Simplify: Combine like terms and simplify each side as you would in an equation.
• Isolate \(x\): Use basic algebraic operations to get \(x\) alone. This includes moving all terms involving \(x\) to one side and constants to the opposite side, just as you saw where \(6x + 18 \geq -1\) was simplified to \(6x \geq -19\).
• Consider the direction: If you multiply or divide by a negative number, reverse the inequality sign. This step is crucial and different from solving equations.

The solution \( x \geq -\frac{19}{6} \) indicates all values of \(x\) that satisfy the inequality. This step-by-step approach ensures you handle each part systematically.