91Ó°ÊÓ

Solving inequalities is very similar to solving equations, but with a few key differences. When you have an inequality like \(x - 3.4 \geq 6.2\), your aim is to find all the values of \(x\) that make this statement true.
To solve the inequality, you need to manipulate it in the same way you would an equation: use basic arithmetic operations such as addition, subtraction, multiplication, or division. However, there is a crucial aspect to remember:

• When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is because the sense of the inequality changes as the number line's direction reverses.

In our original exercise, you only need to add 3.4 to both sides, which doesn't change the inequality's direction. Hence, solving the inequality gives us \(x \geq 9.6\).
This solution tells us that any number 9.6 or larger satisfies the initial statement.

Graphing inequalities on a number line provides a visual representation of the solution set. It helps you instantly see the range of numbers that satisfy the inequality.
To draw this graph:

• Start by drawing a horizontal line and marking numbers in increasing order.
• Find the specific number involved in the inequality, in this case, 9.6.
• Place a solid or closed circle at 9.6 to indicate that this number is included in the solution (because the symbol \(\geq\) includes equality).
• Shade a line or draw an arrow extending to the right side of 9.6 to denote all numbers greater than 9.6 are valid solutions.

Using a number line not only confirms your algebraic solution but also gives a clear picture of how many solutions exist - which, in the case of inequalities, is infinite.

Isolating the variable is one of the primary steps in solving equations and inequalities. The goal is to get the variable by itself on one side of the inequality or equation so that you can clearly see its value.
In the inequality \(x - 3.4 \geq 6.2\), the variable \(x\) is initially not isolated. To remove the 3.4 subtracted from \(x\), you perform the same operation (but opposite) on both sides of the inequality.
Adding 3.4 to both sides effectively cancels out the \(-3.4\) on the left-hand side:

• Left Side: \(x - 3.4 + 3.4 = x\)
• Right Side: \(6.2 + 3.4 = 9.6\)

Thus, you derive \(x \geq 9.6\), where \(x\) is now isolated.
Isolating the variable simplifies the inequality, letting you see the direct relationship between \(x\) and the other numbers involved, meaning you can more easily find a solution.