Problem 19 - Algebra 1: Concepts and Skills

Question

Describe the steps you would use to solve the inequality. $$ 6(z-2)<15 $$

Step-by-Step Solution

Answer

The solution to the inequality is \(z < 4.5\).

Step 1: Distribute

Distribute the 6 to both terms within the parenthesis. This will give \(6*z - 6*2 < 15\). After calculating, we have \(6z -12 < 15.\)

Step 2: Isolate the Term Containing the Variable

To isolate the term with the variable, z, we should add 12 to both sides of the equation. This gives \(6z < 15 + 12\). After adding, we have \(6z < 27.\)

Step 3: Solve for the Variable

To solve for z, divide both sides of the inequality by 6. This gives \(z < 27 / 6\). After dividing, we have \(z < 4.5)\).

Key Concepts

Understanding the Distributive Property in Inequalities

When solving inequalities, especially those with expressions in parentheses, the distributive property becomes very useful. This property is all about simplifying expressions by distributing a single term over addition or subtraction inside the parentheses. For instance, if you see something like \(6(z - 2)\), you apply the distributive property by multiplying the 6 by each term inside the parentheses separately. This results in \(6\cdot z - 6\cdot 2\), which simplifies to \(6z - 12\).
Breaking it down further:

• This technique helps in eliminating parentheses, making it easier to manage the equation or inequality.
• It's crucial to follow the order of operations – multiplication and division before addition and subtraction.

Understanding how to apply the distributive property correctly will simplify many more complex math problems you encounter in the future.

The Art of Variable Isolation

Variable isolation is the essential process of restructuring an equation or inequality to have the variable alone on one side. Once you distribute in our inequality, \(6z - 12 < 15\), your next task is to get \(z\) by itself. Here’s how:

• Add 12 to both sides to counter the subtraction, simplifying our equation to \(6z < 27\).
• Think of it as 'moving' terms from one side of the inequality to the other by performing opposite operations.
• Always perform the same operation to both sides to maintain balance; it’s a bit like a see-saw, keeping it even is the goal!

This method of manipulating equations ensures that the original relationship between terms stays the same, which is crucial for solving accurately.

Mastering Inequality Manipulation

Inequality manipulation involves changing the form of an inequality while keeping its solution set intact. In our example, after isolating the variable term \(6z < 27\), the final step is to solve for \(z\). Here's what to do:

• Divide both sides by the coefficient of \(z\), which is 6, leading to \(z < \frac{27}{6}\).
• Simplify this to \(z < 4.5\).
• Remember: while dividing by or multiplying positive numbers retains the inequality's direction, if you do so by negative numbers, you'll need to reverse the inequality symbol!

Getting comfortable with these manipulations gives you the flexibility to solve various inequality problems with confidence.