Problem 53 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ 3 x+2 \leq 2 x-5 $$

Step-by-Step Solution

Answer

Answer: The solution to the inequality is \(x \leq -7\).

Step 1: Subtract 2x from both sides

We want to move all the x terms to the left side of the inequality. We can achieve this by subtracting 2x from both sides of the inequality. \[ 3x+2 \leq 2x-5 \Longrightarrow (3x-2x)+2 \leq (2x-2x)-5 \]

Step 2: Simplify the inequality

Now we simplify the inequality and obtain a new inequality as follows: \[ x+2 \leq -5 \]

Step 3: Subtract 2 from both sides

Next, we want to isolate x on the left side, which can be done by subtracting 2 from both sides of the inequality: \[ x+2-2 \leq -5-2 \]

Step 4: Simplify and write the solution

Simplifying the inequality, we get the final answer: \[ x \leq -7 \] The solution to the given inequality is \(x \leq -7\).

Key Concepts

Algebraic Inequalities

When we talk about algebraic inequalities, we're discussing the relationship between expressions that are not equal but rather 'less than' or 'greater than.' Instead of an equals sign, we use symbols like <, >, ≤, or ≥ to represent these relationships. Solving algebraic inequalities is much like solving regular equations with the added caveat of paying attention to the inequality direction. For example, if you multiply or divide both sides of an inequality by a negative number, the inequality symbol must be flipped.

It's important to note that the solutions to inequalities are often ranges or sets of values rather than a single number. For instance, in our example above, any value of x that is less than or equal to -7 is a solution to the inequality.

Inequality Simplification

The process of inequality simplification involves minimizing the components of the inequality to make it easier to solve. The primary goal is to boil it down to its most basic form, where the variable we're solving for is on one side, and all the constants are on the other side. To reach this simplified state, you can perform operations such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same number, just as you would with an equation.

It's crucial to carry out these operations carefully to ensure the inequality's direction remains valid. Simplification reduces the inequality step by step, leading to an easier interpretation and solution of the problem, as seen in our exercise where we arrive at the simple inequality x ≤ -7.

Isolate Variable

To isolate the variable means to get the variable by itself on one side of the inequality. This is a fundamental step in solving both equations and inequalities, as it helps to find the solution set for the variable. The process involves using algebraic operations to strategically move all terms other than the variable across the inequality symbol.

For example, if the variable is being added to or subtracted by a number, you would do the opposite operation to both sides to remove that number from the side with the variable. If the variable is being multiplied or divided by a number, you would divide or multiply both sides by that number to isolate the variable. Remember to always preserve the direction of the inequality unless you're multiplying or dividing by a negative number, in which case you must flip the inequality symbol.