Question

Solve the inequality and graph its solution. \(x-5 \geq 1\)

Short answer

The solution for the inequality is \(x \geq 6\) represented by a shaded region on the number line starting at 6 and extending towards infinity.

Step-by-step solution

Isolate x

In order to isolate x, add 5 to both sides of the inequality. This gives the inequality \(x \geq 1 + 5\) or \(x \geq 6\).

Graph the inequality

On a number line, highlight all the points where \(x \geq 6\). That means, starting from 6 (including 6, hence also include a closed circle at point 6 because it's 'greater than or equal to'), shade to the right as it includes all the larger numbers till infinity.

Key concepts

Inequality Graphing

Grasping the technique of inequality graphing is essential for visually understanding how inequalities work. Unlike equations that point out a specific number, inequalities describe a range of possible solutions. To graph an inequality like \(x-5 \geq 1\), you begin by solving the inequality algebraically to find critical values. In this instance, you add 5 to both sides to isolate \(x\), obtaining \(x \geq 6\).

When translating this to a graph, consider a horizontal line which represents a number line. Locate the number 6 and draw a closed circle on it (closed because 6 is included in the solutions, as it's 'equal to or greater than'). From this point, shade the entire line to the right of 6, indicating all greater numbers are part of the solution. It helps to think of the inequality symbol as a direction: \(\geq\) indicates shading to the right, while \(\leq\) would imply shading to the left.

Improvement Advice:

• Be precise when drawing the circle to denote whether it's open or closed, based on whether the variable is strictly greater than (or less than) or equal as well.
• Use consistent shading or patterns to represent the solution set on the number line.
• Clearly label the number line with intervals and ensure that it's long enough to clearly depict the range of solutions.

Algebraic Manipulation

Algebraic manipulation is a cornerstone of solving inequalities, just as it is for solving equations. The aim is to rearrange the inequality to isolate the variable on one side, making the solution clear. This process often involves applying inverse operations to both sides of the inequality. For the given inequality \(x-5 \geq 1\), the solution starts by adding 5 to each side. This operation, known as 'isolating the variable', simplifies the inequality to \(x \geq 6\).

It's important to maintain the balance of the inequality, so remember that whatever you do to one side, you must do to the other. Also, be mindful of the inequality's direction—when you multiply or divide by a negative number, the inequality sign must be flipped.

Improvement Advice:

• Show each step of the algebraic manipulation, clearly explaining the inverse operations used.
• Highlight that the inequality retains its direction when adding/subtracting on both sides but reverses when multiplying/dividing by a negative.
• Reinforce why maintaining the balance (doing the same operation on both sides) is crucial for the validity of the inequality.

Number Line Representation

A number line representation serves as a simple yet powerful visual aid in understanding inequalities. It plots the range of values that satisfy the inequality on a line, giving a quick reference to the solution set. After determining the algebraic solution to the inequality \(x \geq 6\), we represent this solution on a number line.

Place a bold point or a closed circle at 6 on the number line, and extend a shaded region or arrow to the right to show all numbers greater than or equal to 6 are included in the range. The direction of the shade or arrow is crucial, as it depicts the continuation of the inequality's condition—towards greater numbers in this case.

Improvement Advice:

• Make sure that the number line is properly scaled, providing a balanced view of negative and positive values.
• Emphasize the difference between closed and open circles to designate inclusion or exclusion of boundary points.
• Use clear annotations and directional arrows to indicate which direction the inequality extends on the number line.