Problem 42 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality. Then graph the solution set on the real number line. \(2(x+7)-4 \geq 5(x-3)\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x \leq \frac{25}{3}\). This is represented on a number line by a solid point at \(\frac{25}{3}\) with a line stretching to the left indicating values 'less than or equal to' \(\frac{25}{3}\).

Step 1: Simplification

Following the order of operations, all operations in brackets should be done first. This results in the inequality: \(2x+14 - 4 \geq 5x-15\). When simplifying further, the inequality is: \(2x + 10 \geq 5x - 15\).

Step 2: Solve for x

Re-arrange it so that all terms with 'x' are on one side and the numbers are on the other. This gives: \(2x - 5x \geq - 15 - 10\). Further simplification results in: \(-3x \geq -25\). Then, divide both sides by -3. Remember, when dividing or multiplying an inequality by a negative value, the inequality symbol has to be flipped! Thus, the inequality will be: \(x \leq \frac{25}{3}\).

Step 3: Graph the solution

On a number line plot the point \(x = \frac{25}{3}\). This point must be filled in solid because the inequality includes 'equals to.' Draw a line stretching to the left of this point to indicate 'less than or equal to'.

Key Concepts

Graphing Inequalities

Graphing inequalities on a real number line is a visual way to show the range of numbers that satisfy an inequality. When graphing, it's important to consider whether the inequality includes the value itself (indicated by "equals to"). For example, in the inequality \(x \leq \frac{25}{3}\), the \("\leq"\) symbol tells us that \(x\) can be equal to \(\frac{25}{3}\) as well as any value less than it.

• Firstly, mark the number \(\frac{25}{3}\) on the number line. Since the inequality is "less than or equal to," you'll use a solid dot, which indicates that the point itself is part of the solution.
• Secondly, draw an arrow or a line from the marked point extending to the left, covering all the values less than this point showing that all these values are part of the solution set.

This visual representation helps not only to solve inequalities but also to better understand the solution set by showing which numbers make the inequality true.

Algebraic Manipulation

Algebraic manipulation is the process performed to simplify expressions and solve equations and inequalities. It includes adding, subtracting, multiplying, and dividing both sides of an expression or inequality by the same amount to isolate the variable. In the given exercise, the inequality \(2(x+7)-4 \geq 5(x-3)\) was simplified using the following steps:

• First, distribute the numbers outside the brackets: \(2 \times (x + 7) = 2x + 14\) and \(5 \times (x - 3) = 5x - 15\).
• Simplify the expression to \(2x + 10 \geq 5x - 15\) by combining like terms.
• To solve for \(x\), move all variables to one side: subtract \(5x\) from \(2x\) yielding \(-3x\).
• Move the constants to the opposite side by subtracting 10 from both sides, resulting in \(-3x \geq -25\).
• Finally, divide by \(-3\). Remember to flip the inequality sign, yielding \(x \leq \frac{25}{3}\).

Mastering these steps of algebraic manipulation is crucial as they form the foundation for solving more complex inequalities and equations.

Real Number Line

The real number line is a fundamental concept in mathematics, representing all possible numbers that can be plotted along a straight, continuous line. Each point on this line corresponds to a real number. When dealing with inequalities, using a real number line is an effective way to depict the range of possible solutions graphically. For the inequality \(x \leq \frac{25}{3}\), the real number line helps us visualize which numbers are included in the solution. By marking \(\frac{25}{3}\) on the line and shading or drawing a line extending to the negative side, we can easily see all numbers less than or equal to \(\frac{25}{3}\) are part of the solution set. This visual approach simplifies understanding because it allows you to see the complete set, set boundaries, and makes it