Problem 60 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ 2-4 x \leq-3+x $$

Step-by-Step Solution

Answer

Question: Solve the inequality \(2 - 4x \leq -3 + x\), and determine the range of values that satisfy the inequality. Answer: The solution to the inequality is \(x \geq 1\).

Step 1: Add/substract same quantity from both sides

To isolate the variable x, we need to first get all terms with x on one side. In this case, we will begin by adding 4x to both sides of the inequality: $$ 2 - 4x + 4x \leq -3 + x + 4x $$ Simplifying this, we get: $$ 2 \leq -3 +5x $$

Step 2: Add/substract same quantity from both sides (cont.)

Now, we will move the constant term from the right side to the left side. To do this, add 3 to both sides of the inequality: $$ 2 + 3 \leq -3 + 5x + 3 $$ Simplifying this, we get: $$ 5 \leq 5x $$

Step 3: Divide or multiply both sides by the same quantity

To isolate x, we need to divide both sides by 5: $$ \frac{5}{5} \leq \frac{5x}{5} $$ Simplifying this, we get: $$ 1 \leq x $$ Therefore, the solution to the inequality is \(x \geq 1\).

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that helps us represent real-world situations using variables, numbers, and symbols. It enables us to solve problems involving unknown values. In algebra, we work with equations and inequalities to determine the value of variables.
In the given exercise, we are dealing with an inequality. Inequalities are expressions that show how two values relate in size, for example, stating that one value is less than or equal to another. This is expressed with symbols like \(<, >, \leq,\) or \(\geq\).
The equation \(2 - 4x \leq -3 + x\) is an inequality where our goal is to find the values of \(x\) that satisfy it. Algebraic techniques not only help in finding these values but also in visualizing and understanding potential solutions using graphs and number lines. Remember, mastering algebra involves becoming comfortable with manipulating expressions, balancing equations, and reflecting on why each step is necessary.

Variable Isolation

Variable isolation is an essential strategy when solving equations and inequalities. It's about getting the variable of interest alone on one side of the equation or inequality.
In our example, \(2 - 4x \leq -3 + x\), we want to isolate \(x\) to identify its possible values.
Here's how:

• Start by relocating all instances of \(x\) to one side of the inequality to simplify the process.
• This involves performing the same arithmetic operation on both sides to maintain balance. Think of it like balancing a seesaw: whatever you adjust on one side, you need to do the same on the other side.

For example, adding \(4x\) to both sides, we have \(2 - 4x + 4x \leq -3 + x + 4x\) simplifies to \(2 \leq -3 + 5x\).
From here, further isolating \(x\) involves dealing with constants and separating them from the variable, as we show by adding 3 to both sides to get \(5 \leq 5x\). The ultimate step in isolating \(x\) will involve dividing (or multiplying) by the coefficient of \(x\) to leave \(x\) alone.

Simplifying Equations

Simplifying equations is about making expressions easier to work with while retaining their equivalence. This process makes it easier to identify solutions quickly and accurately.
It involves combining like terms and reducing expressions whenever possible.
Taking our inequality example from Step 1, after relocating terms, we had \(2 \leq -3 + 5x\). The next steps included:

• Eliminating the constant on the side with the variable by adding it to both sides, resulting in \(5 \leq 5x\).
• Removing the coefficient of \(x\) by dividing both sides by this coefficient. So, dividing by 5 gives \(1 \leq x\).

This simplification step is crucial because it reduces the complexity of the equation and serves as a foundational skill for solving more complex mathematical problems.
Remember, each simplification step should make sense, maintaining equality or the inequality's nature, and guiding you closer to the solution.