Problem 55 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ 3 x-12 \geq 7 x+4 $$

Step-by-Step Solution

Answer

Question: Solve the inequality and find the values of x for which the inequality holds: 3x - 12 ≥ 7x + 4. Answer: x ≤ -4.

Step 1: Simplify the inequality

To simplify the inequality, we want to get all the terms with x on the same side of the inequality sign. To do this, we'll add -7x to both sides of the inequality: $$ 3x - 7x - 12 \geq 7x - 7x + 4 $$ Now, simplify both sides of the inequality: $$ -4x - 12 \geq 4 $$

Step 2: Isolate x

Now, we want to isolate x on the left side of the inequality. First, we'll add 12 to both sides of the inequality: $$ -4x - 12 + 12 \geq 4 + 12 $$ Simplifying, we get: $$ -4x \geq 16 $$ Next, divide both sides of the inequality by -4 to get x by itself. Remember that when we divide or multiply an inequality by a negative number, we need to reverse the inequality sign: $$ \frac{-4x}{-4} \leq \frac{16}{-4} $$ Simplifying, we get: $$ x \leq -4 $$

Step 3: Check the inequality

Now that we've solved for x, let's check the inequality with some example values to make sure it is true. Suppose x = -5, then the left side of the original inequality: $$ 3(-5) - 12 = -15 - 12 = -27 $$ The right side of the original inequality: $$ 7(-5) + 4 = -35 + 4 = -31 $$ Since -27 ≥ -31, the inequality holds. Therefore, our solution is correct: $$ x \leq -4 $$

Key Concepts

Algebraic Inequalities

Algebraic inequalities are mathematical expressions that show the relationship between two values where one is not necessarily equal to the other, but rather greater than or less than. Understanding inequalities is crucial because they appear in a variety of real-world situations, such as determining speed limits, minimum age requirements, and financial budget constraints.

When solving an algebraic inequality, such as \( 3x - 12 \geq 7x + 4 \), our goal is to find all the values of the variable that make the inequality true. To achieve this, we perform similar operations as we do when solving equations, with the added attention to how multiplying or dividing by negative numbers affects the inequality's direction.

Isolate Variables

Isolating the variable in an inequality is a similar process to solving an equation. The aim is to get the variable by itself on one side of the inequality so that we can see the range of values it can take. For example, {'text': in case you are making a solution involving subtraction or addition, ensure you maintain balance by doing the same to both sides---if you add a number to one side, you have to add the same number to the other side, as we do with \(-4x - 12 + 12 \geq 4 + 12\).

This step gets us closer to finding the solution set for the variable. It is the simplification process that separates the variable from the constants by using the properties of addition, subtraction, multiplication, and division. Once the variable is isolated, as in \(-4x \geq 16\), we can clearly see the critical step needed to solve the inequality—division by the coefficient of the variable.

Inequality Sign Reversal

Inequality sign reversal is a key concept to remember when solving inequalities. Unlike equations, when you multiply or divide both sides of an inequality by a negative number, the inequality sign must be flipped. This is because reversing the sign ensures the inequality's truth is maintained throughout the operation.

For example, divіdіng both sides of \(-4x \geq 16\) by -4, a negative number, reverses the inequality to become \(x \leq -4\). Failure to reverse the sign would result in an incorrect solution set. Therefore, a solid understanding of when and why to reverse the sign is essential for finding the correct solutions to inequalities.