Question

Solve the inequality. Graph the solution on a number line. \(4 x-4>8\)

Short answer

The solution for the inequality \(4x - 4 > 8\) is \(x > 3\). On a number line, an open circle would be placed on 3 and a line will be drawn to the right to indicate all numbers greater than 3.

Step-by-step solution

Simplify the Inequality

The first step is to simplify the inequality by getting rid of the number -4 from the left side. This is done by adding 4 on both sides of the inequality: \(4x - 4 + 4 > 8 + 4\), simplifying, you get: \(4x > 12\).

Solve for x

The next step is to isolate \(x\). This is done by dividing all terms by 4: \(4x / 4 > 12 / 4\). Simplifying, the solution for \(x\) will be \(x > 3\).

Graph the Solution

On a number line, plot a point at the number 3, and because \(x\) is 'greater than' and not 'greater than or equal to', use an open circle at this point. Then draw an arrow to the right of 3, which indicates that the solution includes all numbers greater than 3.

Key concepts

Inequality Graphing

Inequality graphing is a visual way to represent the solutions to an inequality. When you have an inequality such as the one given, \(4x - 4 > 8\), it's important to first solve for the variable before graphing. Once simplified, as seen with \(x > 3\), the graphing process begins.

To graph \(x > 3\), a number line is used. Identify the number 3 on the line and place an open circle over it to signify that 3 is not part of the solution since \(x\) is strictly greater than 3, not equal to it. From there, draw an arrow to the right, indicating all numbers greater than 3 fulfill the inequality. This arrow shows the range of values for \(x\) that makes the inequality true.

Graphing inequalities allows you to visually confirm the range of possible solutions and is a vital step in understanding inequalities in algebra.

Algebraic Manipulation

Algebraic manipulation involves the strategic use of algebraic operations to solve equations or inequalities. The goal is to isolate the variable of interest, here \(x\), on one side of the inequality. In the exercise given, the manipulation starts by balancing out the -4 on the left by adding 4 to both sides, resulting in \(4x - 4 + 4 > 8 + 4\), which simplifies to \(4x > 12\).

Following that, divide both sides by 4 to find the value of \(x\), so \(4x / 4 > 12 / 4\) simplifies to \(x > 3\). It's crucial to preserve the balance of the inequality, similar to an equation, during the manipulation process. Remember, for inequalities, if you multiply or divide by a negative number, the inequality direction changes, which didn't happen here since we divided by positive 4.

Number Line Representation

The number line is an essential tool for representing and visualizing real numbers, including solutions to inequalities. In the context of the given inequality, after algebraic manipulation reveals that \(x > 3\), the number line helps to illustrate this range of solutions.

To accurately represent \(x > 3\), you place an open circle on the number line at the 3 to indicate that 3 itself is not included. Then, an arrow extending to the right of 3 shows all the numbers that \(x\) can be, as they are all greater than 3. This clear, visual representation ensures that anyone who looks at the number line immediately understands the set of all possible solutions to the inequality.