Problem 41 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The solution is \(x < -18\). It is represented on the number line where the x-values less than -18 are shaded. The point at -18 on the number line is open indicating that -18 is not part of the solution.

Step 1: Distribute the multiplication on the left side

Start by distributing the 3 on left side across the brackets.\nSo, \(3(x+2)+7 < 2x -5\) transforms to \( 3x + 6 + 7 < 2x - 5)\

Step 2: Simplify both sides

Sum up the positive constants on the left side and simplify right side.\nWe get \(3x + 13 < 2x -5\) .

Step 3: Isolate x

By subtracting 2x from both sides and then subtracting 13, we isolate x, this gives us:\n\(x < -18\)

Step 4: Representation on Number Line

Draw a number line and mark -18 on it. The values of x that are less than -18 are shaded on the number line. The circle at -18 is open indicating that -18 is not included in the solution set.

Key Concepts

Real Number Line

The real number line is a visual representation of all the real numbers. It stretches infinitely in both directions, much like a straight road without end. The center is usually denoted by 0. Every point to the left of 0 represents negative numbers, and every point to the right represents positive numbers.
When graphing a solution set on the real number line, like in our given inequality, we use open or closed circles to indicate whether endpoints are included in the set. An open circle means that the number at that point isn’t included, while a closed circle means it is. For example, in the inequality solution \(x < -18\), an open circle is placed at -18, and all numbers to the left are shaded.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. It’s about finding the unknown or putting real-life variables into equations and then solving them.
In algebra, we often deal with expressions and equations. An expression is a combination of numbers, variables (like x or y), and operations (like addition or multiplication). An equation states that two expressions are equal, and our task is to find the value of variables that make it true.

• For example, in \(3x + 13 < 2x - 5\), we deal with the inequality using basic algebraic principles like combining like terms and isolating variables.

Solution Set

A solution set is the collection of all possible solutions that satisfy an equation or inequality.
In the context of inequalities, the solution set often includes a range of values. These values represent everything that meets the condition given by the inequality. For the inequality \(x < -18\), the solution set includes all real numbers less than -18.

• You represent this on the real number line by shading everything to the left of -18 and using an open circle at -18.
• This visual representation helps in quickly understanding the range of values x can take to make the inequality true.

Problem Solving

Problem solving involves breaking down a complex problem into simpler parts to find a solution.
Mathematics, especially algebra, relies heavily on problem solving. When solving an inequality:

• Start by simplifying each side of the inequality, as is done when distributing \(3(x+2) + 7\) in our example.
• Next, isolate the variable on one side using appropriate algebraic operations such as adding, subtracting, and dividing.
• Finally, graph the solution to visually confirm your work, as shown by marking on the real number line.

This structured approach ensures that you arrive at the correct solution efficiently and clearly.