Question

Solve the inequality. \(2.1 x \leq-10.5\)

Short answer

So, for the inequality to hold true, \(x\) must be greater or equal to -5. Thus, \(x \geq -5\).

Step-by-step solution

Analyze the inequality

The given inequality is \(2.1x \leq -10.5\). The aim is to isolate 'x' because we want to find the value for 'x' that satisfies this inequality.

Divide both sides by the coefficient of x

To isolate 'x', we need to remove the coefficient of 'x' which is '2.1'. We can do this by dividing both sides of the inequality by '2.1'. So the operation becomes, \(- \frac{10.5}{2.1} \leq x\).

Solve the operation

Solving the operation \(- \frac{10.5}{2.1}\) gives a result of '-5'. This means that 'x' is much greater or equal to '-5' to satisfy the inequality, hence, \(x \geq -5\).

Key concepts

Algebraic Inequalities

When we talk about algebraic inequalities, we're referring to mathematical statements that compare expressions where equality isn't necessarily present. Instead of an '=' sign, you'll see symbols like '\textless', '\textgreater', '\textless=', and '\textgreater='. Just like in the exercise \(2.1 x \leq-10.5\), these symbols show the relationship between two expressions. To solve them, the process often involves performing similar operations as you would with an equation, but with careful attention to the rules of inequality. For example, when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This ensures that the statement's truth value remains unchanged.
Understanding and solving algebraic inequalities is essential because it allows you to identify a range of possible solutions, which is particularly valuable in real-life situations where conditions are variable and not fixed.

Isolate Variable

In algebra, isolating the variable is a key step in solving equations and inequalities; it means to get the variable you're solving for by itself on one side of the equation or inequality. This is achieved through various algebraic manipulations such as addition, subtraction, multiplication, division, and application of inverse operations. In our exercise, to isolate 'x' from \(2.1x \leq -10.5\), we're trying to get 'x' alone on one side. This involves dividing both sides by the coefficient of 'x', which is '2.1'. The goal is to have an inequality that reads something like 'x \textless number', 'x \textgreater number', 'x \leq number', or 'x \geq number'. By isolating the variable, the range of all possible values that satisfy the inequality becomes clear, making these solutions easier to understand and visualize.

Division Property of Inequality

The division property of inequality states that when you divide both sides of an inequality by a positive number, you keep the inequality sign as it is. However, when dividing by a negative number, you must reverse the inequality sign to maintain the inequality's truthfulness. This property is essential when working with inequalities because it ensures the solution set remains correct after the division operation.
In the example \(2.1x \leq -10.5\), we apply this property by dividing both sides of the inequality by '2.1', a positive number. This operation maintains the direction of the inequality and simplifies it to \(x \geq -5\). This step is straightforward, but always remember, division by a negative number would flip the inequality sign, a mistake that some students make. The division property of inequality helps students solve for the variable accurately without altering the original inequality's meaning.