Problem 37 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The solution to the inequality is \(x>2\). On a number line, this is represented by an open circle at 2 and an arrow to the right.

Step 1: Simplify the inequality

Subtract \(2x\) from both sides of the inequality to isolate the variable term on one side, this gives you \(7<3+2x\).

Step 2: Further simplify the inequality

Now subtract 3 from both sides of the inequality to get \(4<2x\).

Step 3: Solve for the variable

Finally, divide both sides by 2 to solve for \(x\), which gives you \(x>2\).

Step 4: Graph the solution on the number line

The inequality \(x>2\) means that all real numbers greater than 2 are part of the solution set. On a number line, this is represented by an open circle at 2 (as 2 is not part of the solution) and an arrow going to the right indicating all numbers greater than 2 are part of the solution.

Key Concepts

Graphing Inequalities

Graphing inequalities on a number line helps you visualize solutions. When you graph an inequality like \(x > 2\), it tells a story about all the numbers that meet the condition. The \(x > 2\) inequality means any number larger than 2 will qualify. Envision your number line as a road test for this statement. You place an open circle on 2 to show it's the start, not included. Then, draw an arrow to the right. The arrow implies that the numbers running along the road, to the right of the circle, fulfill \(x > 2\).
An open circle signifies that the starting number in the inequality (which is 2 here) is not included. This stands in contrast to \(x \geq 2\), which would include 2, represented by a filled dot. The direction of your arrow depends on the inequality: for greater-than, point right; for less-than, point left. Through this visual process, solving inequalities becomes an activity of clarity rather than mere algebra.

Real Number Line

The real number line is a straight line that extends infinitely in both directions, symbolizing the range of real numbers. Each position on this line corresponds to a real number, allowing us to plot solutions to equations and inequalities effectively. This lines up every real number sequentially, with negative numbers to the left of zero and positive numbers to the right.
In the context of our inequality \(x > 2\), any point right of 2 aligns with all numbers greater than 2. This is easily depicted on our real number line. Armed with this visual tool, identifying solutions to inequalities transforms from theoretical understanding to tangible observation. Just remember:

• Each point represents a unique real number.
• The line continues endlessly in both directions.
• Periodic branching arrows (or dots) visually track possible solutions to inequalities.

This concept is a keystone in understanding not only inequalities but also broader mathematical equations.

Algebraic Manipulation

Algebraic manipulation involves the strategic rearrangement of equations to isolate a variable. In solving inequations such as \(2x + 7 < 3 + 4x\), these techniques become invaluable. Here's how:

• First, simplify by removing like terms from one side — subtract \(2x\) from both sides to yield \(7 < 3 + 2x\).
• Then, further simplify by clearing constants next to the variable — subtract 3 to get \(4 < 2x\).
• Finally, isolate the variable by dividing, providing a neat result \(x > 2\).

The focus of algebraic manipulation is simplification and clarity. Each step in the inequality's transformation brings you closer to a clean, understandable outcome, ready for graphical or numerical interpretation. At its core, algebraic manipulation is about clarity and control, letting you mould complex expressions into simpler, easily solvable ones.