Problem 100 - Introductory Algebra for College Students

Question

Solve: \(\quad 8(x-2)-2(x-3) \leq 8 x\).

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq -5\)

Step 1: Distribute the multiples across the brackets

We start by distributing the 8 and -2 across the brackets: \(8x - 16 - 2x + 6 \leq 8x\).

Step 2: Combine like terms

We combine the like terms on the left side of the inequality gets: \(6x - 10 \leq 8x\).

Step 3: Isolate x

We can subtract 6x from both sides which makes the inequality: \(-10 \leq 2x\).

Step 4: Solve for x

Lastly, we divide both sides by 2 to find that \(x \geq -5\)

Key Concepts

Inequality Solving

Solving inequalities is similar to solving equations, but there are a few additional rules to keep in mind. Inequalities express a comparison between two values. When we solve an inequality, we find the set of values that satisfy the inequality condition.

• Inequalities can be less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥).
• The goal is to isolate the variable just like in equations, but when multiplying or dividing by a negative, flip the inequality sign.
• Once solved, inequalities are often represented on a number line or with interval notation.

Understanding these basic principles will make solving more complex inequalities much easier. Always think about the inequality like a balance; whatever you do to one side, you must do to the other.

Distributive Property

The distributive property is a key algebraic property used to simplify expressions and is essential for solving equations and inequalities. It states that multiplying a single term by terms inside a parenthesis can be expressed as distributing the multiplication to each term inside. In the exercise, this principle is applied as:

• The term 8 multiplies each of the terms within the parentheses: \(8(x-2) = 8x - 16\).
• Similarly, \(-2(x-3)\) becomes \(-2x + 6\).

This step simplifies expressions, making it easier to manage and solve inequalities or equations. Knowing when and how to apply the distributive property is crucial for simplifying algebraic expressions.

Combining Like Terms

Combining like terms is a process in algebra that simplifies expressions to fewer terms, making it more manageable. Like terms are terms that contain the same variable raised to the same power. To combine them, you simply add or subtract their coefficients. In the solution:

• We had \(8x - 16\) and \(-2x + 6\) from applying the distributive property.
• Combined terms are \(8x - 2x = 6x\) and the constants \(-16 + 6 = -10\).

This results in the expression becoming \(6x - 10\). Streamlining expressions in this way sets the stage for more straightforward problem solving and helps in further isolation of the variable.

Solving Linear Inequalities

Solving linear inequalities involves finding all the possible values for the variable that make the inequality statement true. Once you simplify the inequality through distributing and combining terms, you can isolate the variable by using inverse operations. In the step-by-step solution:

• After combining terms, the inequality becomes \(6x - 10 \leq 8x\).
• We isolate \(x\) by subtracting \(6x\) from both sides: \(-10 \leq 2x\).
• Finally, divide both sides by 2: \(-5 \geq x\), or rewritten as \(x \geq -5\).

These steps make sure the inequality is maintained through logical manipulations, leading to a solution set for the variable. Always check each step to make sure rules specific to inequalities are followed, such as flipping the inequality when multiplying or dividing by a negative.