Problem 44 - Understanding Elementary Algebra with Geometry

Question

Solve the given inequality and sketch the solution set on a number line. $$4 x+7 \leq 6 x+17$$

Step-by-Step Solution

Answer

Solution: \(x \geq -5\). Shade the number line to the right of and including \(-5\).

Step 1: Simplify the inequality

Subtract \(4x\) from both sides to get \(7 \leq 2x + 17\).

Step 2: Isolate the variable term

Subtract \(17\) from both sides to obtain \(7 - 17 \leq 2x\). This simplifies to \(-10 \leq 2x\).

Step 3: Solve for the variable

Divide both sides by \(2\) to isolate \(x\). This yields \(-5 \leq x\). So the solution is \(x \geq -5\).

Step 4: Sketch the solution on the number line

Draw a number line and shade all numbers greater than or equal to \(-5\). Make a solid point at \(-5\) and shade to the right.

Key Concepts

Variable Isolation

To solve inequalities, it's crucial to isolate the variable. This means we need to get the variable alone on one side of the inequality sign. In the exercise, we start with: \(4x + 7 \leq 6x + 17\). By subtracting \(4x\) from both sides, we work towards isolating the variable term \(x\). This leaves us with: \(7 \leq 2x + 17\). Remember, whatever you do to one side, you must also do to the other. This helps maintain the balance of the inequality.

Simplifying Inequalities

Simplifying inequalities involves performing arithmetic operations to make the inequality simpler and more understandable. Following from our previous step where we had: \(7 \leq 2x + 17\), we subtract \(17\) from both sides to further simplify: \(7 - 17 \leq 2x\), which becomes \(-10 \leq 2x\). This step reduces the inequality to a simpler form, making it easier to solve. Simplifying also eliminates any unnecessary terms and focuses on the main components involving the variable \(x\).

Number Line Representation

Visualizing the solution to an inequality can be very helpful. After isolating the variable to obtain \(x \geq -5\), we can sketch this solution on a number line. Draw a straight line and mark important points. Place a solid point at \(-5\) since the inequality includes \(-5\) itself (\(\geq\) indicates greater than or *equal* to). Then, shade the line to the right of \(-5\) to represent all numbers that are greater than or equal to \(-5\). This shaded region visually represents the set of all possible solutions to the inequality.

Linear Inequalities

Linear inequalities, like the one given in our exercise, involve inequalities with linear expressions on both sides. These expressions may involve constants and the first power of the variable. The goal is to manipulate these expressions, using addition, subtraction, multiplication, or division, to isolate the variable. For example, in \(4x + 7 \leq 6x + 17\), we performed arithmetic operations to simplify it to \(x \geq -5\). Linear inequalities can have one solution, no solution, or an infinite number of solutions, represented on the number line. This exercise demonstrates how simplifying and isolating terms are key steps that lead to a clearer understanding of the solution.