Problem 96 - Elementary Algebra

Question

Solve the inequalities. $$ 5 x+4 \geq 7 x+16 $$

Step-by-Step Solution

Answer

Question: Solve the inequality and find the solution for x: $$5x + 4 \geq 7x + 16$$ Answer: The solution for the given inequality is $$x \leq -4$$.

Step 1: Write down the given inequality

The given inequality is: $$ 5x + 4 \geq 7x + 16 $$

Step 2: Bring the x terms to one side and the constant terms to the other side

To do this, we'll subtract 5x from both sides and subtract 4 from both sides: $$ 5x - 5x + 4 \geq 7x - 5x + 16 -4 $$

Step 3: Simplify and solve for x

After simplifying the inequality, we get: $$ 4 \geq 2x + 12 $$ Now, we'll subtract 12 from both sides to isolate the x term: $$ 4 - 12 \geq 2x $$ Simplifying, we get: $$ -8 \geq 2x $$ Now, we'll divide both sides by 2 to solve for x: $$ -4 \geq x $$ The solution of the inequality is: $$ x \leq -4 $$

Key Concepts

Algebraic Inequalities

Understanding algebraic inequalities is foundational for tackling a variety of mathematical problems. An inequality is like an equation, but instead of saying two things are exactly equal, it tells us that one quantity is greater or less than another. These can be simple, such as 2 > 1, or more complex involving variables, as seen in the exercise you're working on.

When solving algebraic inequalities, it's important to remember that the properties of inequality are similar to those of an equation. You can add, subtract, multiply, or divide both sides of the inequality by the same number. However, a crucial difference is when you multiply or divide by a negative number, the inequality sign flips.

For instance, if you have -2x > 6 and you want to solve for x, dividing both sides by -2 would flip the greater than sign to less than, resulting in x < -3. This pivotal rule ensures the relationship between the numbers remains accurate.

Inequality Simplification

A key to solving inequalities efficiently is simplification. This process makes the inequality more manageable and easier to solve. Simplification might involve combining like terms, distributing, or getting variables to one side and numbers to the other, as you've seen in the textbook exercise.

In simplifying the inequality from the exercise, the steps included subtracting the terms containing the variable x from both sides to get all the x terms on one side and the constants on the other. Next, by eliminating zero pairs and combining like terms, the inequality was reduced to a more straightforward form. Every simplification action should help make the inequality clearer, leading to an easier identification of the solution set.

Always be methodical: change one thing at a time, and ensure you do the same thing to both sides of the inequality. This systematic approach keeps the inequality balanced and leads to accurate solutions.

Linear Inequalities

Linear inequalities are expressions that can be simplified to the form ax + b > c or ax + b < c, where x is a variable, a, b, and c are constants, and the inequality sign can also be ≥ or ≤. The example from your homework is a linear inequality because the highest power of x is 1.

Solving a linear inequality closely mirrors the process of solving a linear equation. You perform the same operations to both sides of the inequality to isolate the variable and find its solution or range of solutions. The main goal is to get the variable by itself on one side of the inequality sign. Once you've isolated the variable, you can interpret the inequality: for instance, x ≤ -4 means that x can be any number less than or equal to -4.

Representing the solution graphically on a number line can provide a visual understanding of all the possible values x can take, making linear inequalities a powerful tool in solving real-world problems where a range of solutions is applicable.