Problem 46 - Algebra and Trigonometry

Question

In all exercises, other than \(\varnothing,\) use interval notation to express solution sets and graph each solution set on a number line. In Exercises \(27-50,\) solve each linear inequality. $$ \frac{4 x-3}{6}+2 \geq \frac{2 x-1}{12} $$

Step-by-Step Solution

Answer

The solution to the given inequality is \(x \geq - \frac{19}{6}\), and this is represented in interval notation as \([- \frac{19}{6}, +\infty)\].

Step 1: Clear the Denominators

Start by multiplying everything in the inequality by 12 (which is the lcm of 6 and 12) to get rid of the denominators: \(12[ \frac{4x - 3}{6} + 2 ] \geq 12[ \frac{2x - 1}{12} ]\). This simplifies to: \(8x - 6 + 24 \geq 2x - 1\).

Step 2: Collect Like Terms

Next, combine like terms: \(8x + 18 \geq 2x - 1\). Then get all the x-terms on one side and the constant terms on the other: \(8x-2x \geq -1 - 18\).

Step 3: Isolate the Variable

Continue the simplification: \(6x \geq -19\). Divide by 6 to isolate x: \(x \geq - \frac{19}{6}\).

Step 4: Express in Interval Notation and Graph the Solution

The solution in interval notation is: \[[- \frac{19}{6}, +\infty)\]. On the number line, make a closed dot at \(x = - \frac{19}{6}\) and shade towards right indicating all numbers greater than or equal to \(- \frac{19}{6}\) are the solutions.

Key Concepts

Interval Notation

Interval notation is a concise way of expressing a set of numbers between two endpoints. When we have a range of numbers that make an inequality true, interval notation allows us to express this range clearly and simply. An interval can be bounded or unbounded. For example, when we write \( [a, b] \), it includes all numbers between \( a \) and \( b \), including the endpoints \( a \) and \( b \). In contrast, \( (a, b) \) would include all numbers between \( a \) and \( b \), but not \( a \) or \( b \) themselves.

Inequality Solutions

Solving linear inequalities is similar to solving linear equations, but with an extra consideration for the direction of the inequality. The solution to a linear inequality is not just a single number, but a range of possible values that satisfy the inequality. It's essential to remember that multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol. In the case of \( x \geq - \frac{19}{6} \), the solution set includes all values of \( x \) that are greater than or equal to \( - \frac{19}{6} \) .

Number Line Graphing

Graphing the solution set of an inequality on a number line helps to visually represent the range of values that satisfy the inequality. To graph \( x \geq - \frac{19}{6} \) on a number line, you would draw a closed circle at \( - \frac{19}{6} \) to indicate that this number is included in the solution set, and then shade the line to the right of this point. This shaded area represents all the numbers that are greater than or equal to \( - \frac{19}{6} \) .