Problem 65 - Algebra 2

Question

Solve each inequality. $$ 2 z+5 \geq 1475 $$

Step-by-Step Solution

Answer

\(z \geq 735\)

Step 1: Isolate the variable term

Start by isolating the term with the variable on one side of the inequality. We have the inequality \(2z + 5 \geq 1475\). To isolate \(2z\), subtract 5 from both sides:\[2z + 5 - 5 \geq 1475 - 5\]This simplifies to:\[2z \geq 1470\]

Step 2: Solve for the variable

Now we need to solve for \(z\). Since \(2z\) means 2 times \(z\), divide both sides of the inequality by 2:\[\frac{2z}{2} \geq \frac{1470}{2}\]This reduces to:\[z \geq 735\]

Key Concepts

Isolating the Variable Term

In algebra, isolating the variable term is often the first step in solving inequalities or equations. This means getting the variable with its coefficient on one side of the inequality, while putting all constant numbers on the other side. To do this, you may need to use basic arithmetic operations like addition, subtraction, multiplication, or division.
For example, in the inequality \(2z + 5 \geq 1475\), the term with the variable is \(2z\). To isolate \(2z\), we subtract 5 from both sides.

• The subtraction is valid and keeps the inequality true if you do it to both sides.
• This results in the inequality \(2z \geq 1470\).

Once the variable term is isolated, you're one step closer to solving the inequality.

Two-Step Inequalities

Two-step inequalities, like the example \(2z + 5 \geq 1475\), require two separate mathematical operations to find the solution for the variable.
1. **First Step:** Isolate the variable term using addition or subtraction. In our case, we subtracted 5 to get \(2z \geq 1470\).
2. **Second Step:** Solve for the variable using multiplication or division. Now, divide both sides by 2, giving \(z \geq 735\).
Each step aims to maintain the inequality's balance, meaning whatever operation is applied to one side must be applied to the other.
Understanding these two steps simplifies complex problems and makes solving inequalities straightforward.

Algebraic Solutions

An algebraic solution involves solving a problem using algebraic methods and formulas. When dealing with inequalities like \(2z + 5 \geq 1475\), these solutions are systematic and clear.

• The initial goal is to simplify and manipulate the inequality into a form where the variable is easy to solve.
• Using operations that preserve the inequality's order is crucial, especially when multiplying or dividing by negative numbers, which can flip the inequality sign.
• The aim is to find the set of values that satisfy the inequality, represented in a solution like \(z \geq 735\).

By grasping algebraic solutions, you can solve any inequality by following logical and methodical steps to arrive at the correct outcome.