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Chapter 3: Problem 19

Solve each inequality. $$2 x-9 \geq 16$$

Short Answer

Expert verified
x \geq 12.5

Step by step solution

01

Isolate the variable term

Start by isolating the term with the variable. Add 9 to both sides of the inequality: \[ 2x - 9 + 9 \geq 16 + 9 \] This simplifies to: \[ 2x \geq 25 \]
02

Solve for the variable

Now, solve for the variable by dividing both sides of the inequality by 2: \[ \frac{2x}{2} \geq \frac{25}{2} \] This simplifies to: \[ x \geq \frac{25}{2} \]
03

Write the final answer

The solution to the inequality is: \[ x \geq \frac{25}{2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

inequality
An inequality is a mathematical statement that compares two expressions and shows that one is either less than, greater than, less than or equal to, or greater than or equal to the other. Unlike equations, which state that two expressions are exactly equal, inequalities tell us that there is a range of possible values. For example, the inequality \(2x - 9 \geq 16\) states that the expression on the left is greater than or equal to 16. Inequalities are useful in many real-world situations, like determining minimum requirements or maximum capacities.
isolate variable
To solve an inequality, we first need to isolate the variable on one side of the inequality. Isolating the variable means getting it by itself, which helps us understand the range of values it can take. In our example, \(2x - 9 \geq 16\), we start by isolating the term containing the variable \(2x\). To do this, we add 9 to both sides:
\(2x - 9 + 9 \geq 16 + 9\). Adding 9 cancels out the \-9\ generating \(2x \geq 25\). Now that we have isolated \(2x\), we can proceed to solving for the variable.
solve for variable
Once the variable is isolated, the next step is to solve for the variable. Solving for the variable means finding the specific value or range of values that satisfy the inequality. For the inequality \(2x \geq 25\), we solve for x by dividing both sides by 2:
\(\frac{2x}{2} \geq \frac{25}{2}\).
This results in:
\(x \geq \frac{25}{2}\).
Make sure to keep the inequality sign the same while performing operations such as addition, subtraction, multiplication, or division. It's also important to note that if we ever multiply or divide by a negative number, we must reverse the inequality sign.
final solution
After isolating the variable and solving for it, we can state the final solution. The final solution to the inequality \(2x - 9 \geq 16\) is:
\(x \geq \frac{25}{2}\).
This means that x can be any value greater than or equal to \frac{25}{2}\, which simplifies to 12.5. We can represent this solution graphically on a number line, showing all the values that x can take. In real-life applications, this helps us understand the possible and acceptable values for certain conditions or constraints.

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