Problem 11 - Prealgebra

Question

Solve each inequality. Check your solution. $$13 a \geq-26$$

Step-by-Step Solution

Answer

Solution: \(a \geq -2\). Check: True for \(a = -2\) and values greater.

Step 1: Isolate the variable

To solve the inequality \(13a \geq -26\), we need to isolate \(a\). We do this by dividing both sides of the inequality by 13. This gives us: \[ a \geq \frac{-26}{13} \] Perform the division: \[ a \geq -2 \] This is the solution to the inequality.

Step 2: Check the solution

We need to verify that \(a = -2\) is part of the solution and check for values greater than \(a\). Substitute \(a = -2\) into the original inequality: \[ 13(-2) = -26 \] Since \(-26 \geq -26\) is true, \(a = -2\) is a solution.Now, check a value greater than \(-2\), such as \(a = 0\):\[ 13(0) = 0 \] Since \(0 \geq -26\) is true, any value \(a\geq-2\) satisfies the inequality.

Key Concepts

Solving Inequalities

Solving inequalities involves finding all the possible values of a variable that satisfy the given inequality. Unlike solving equations, when solving inequalities, we look at a range of values rather than just one particular value. Here's how you can approach solving them:

First, determine the mathematical operation required to isolate the variable. In the given exercise, to solve the inequality \(13a \geq -26\), you need to perform the opposite operation of multiplication, which is division.
Divide each side of the inequality by 13 to get \(a \geq \frac{-26}{13}\).

Performing the division results in \(a \geq -2\). This means that any number equal to or greater than -2 will satisfy the inequality.

Remember these key points when solving inequalities:

• When dividing or multiplying by a negative number, reverse the inequality sign.
• Find the simplest form of the inequality first before proceeding.
• Ensure all variable terms are collected on one side of the inequality.

Checking Solutions

Once an inequality is solved, it's crucial to verify whether the solution set is accurate. Checking solutions involves substituting values back into the original inequality to see if they satisfy the condition.

In our example, by substituting \(a = -2\) into the inequality \(13a \geq -26\), we compute \(13(-2) = -26\).
Since \(-26 \geq -26\) is true, \(a = -2\) is indeed a correct part of the solution.

To be thorough, test a value greater than \(-2\). Substitute \(a = 0\) and find: \(13(0) = 0\).
As \(0 \geq -26\) holds true, the solution \(a \geq -2\) is confirmed.

When checking solutions, keep these tips in mind:

• Always test multiple values from the solution set to ensure accuracy.
• Start with the boundary condition of the inequality and proceed with values that are clearly within the solution set.

Isolation of Variables

Isolation of variables is a technique used to simplify equations or inequalities and make them easier to solve. The main goal is to have the variable by itself on one side of the equation or inequality.

In order to isolate a variable in inequalities like \(13a \geq -26\), you want to perform operations that will leave the variable by itself.

Divide each side by the coefficient of the variable (13 in this case) to isolate \(a\).
This results in \(a \geq \frac{-26}{13}\), simplifying further to \(a \geq -2\).

Follow these steps to effectively isolate variables:

• Use inverse operations—such as addition or subtraction—to move constants away from the variable.
• Divide or multiply to get the variable alone.
• Always perform the same operation on both sides of the inequality to maintain balance.

By mastering this skill, tackling inequalities becomes a straightforward task, paving the way for successfully solving more complex mathematical problems.