Problem 50 - Prealgebra

Question

Solve each inequality. Check your solution. $$22 \leq-15+y$$

Step-by-Step Solution

Answer

The solution is \(y \geq 37\).

Step 1: Isolate the Variable on One Side

To solve the inequality \(22 \leq -15 + y\), we need to isolate \(y\). To do this, we add 15 to both sides of the inequality. This gives us: \[ 22 + 15 \leq y \].

Step 2: Simplify the Expression

Now, simplify the left side of the inequality: \[ 22 + 15 = 37 \]. Therefore, we now have \(37 \leq y\). This can be rewritten as \(y \geq 37\).

Step 3: Check the Solution

To ensure the solution is correct, substitute \(y = 37\) back into the original inequality \(22 \leq -15 + y\). Plugging in 37 gives us \(22 \leq -15 + 37\). Simplifying the right side we get: \(-15 + 37 = 22\), thus \(22 \leq 22\), which is true. Therefore, \(y = 37\) is a correct boundary value of our solution set.

Key Concepts

Solving Inequalities

In prealgebra, solving inequalities is similar to solving regular equations, with a few important differences. An inequality is a mathematical statement that compares the sizes or values of two expressions using inequality signs such as \(<, \leq, >, \geq\). The goal when solving them is to determine the set of values that satisfy the inequality, rather than just a single solution.

To solve an inequality like \(22 \leq -15 + y\), begin by treating it as though it’s an equation where we solve for the variable. This will involve performing operations to both sides of the inequality sign in order to isolate the variable, without changing the original sense of the inequality. Remember to:

• Add or subtract the same number on both sides.
• Multiply or divide both sides by a positive number.

This is exactly what we do in our example: adding 15 to both sides, leading us to \(37 \leq y\). Simplifying further gives us the equivalent inequality \(y \geq 37\), showcasing the set of possible solutions where \(y\) is all values greater than or equal to 37.

Checking Solutions

Verifying the accuracy of your solution in inequalities is as important as finding it. Once you believe you've determined the correct inequality, it's crucial to substitute values from your solution back into the original inequality. This helps ensure that your solution holds true.

For example, after we found that \(y \geq 37\), it's important to check this by plugging \(y = 37\) back into the original inequality: \(22 \leq -15 + 37\). By simplifying \(-15 + 37\), you obtain 22, confirming \(22 \leq 22\), a true statement.

When checking inequalities, always test the boundary value (here \(y = 37\)) and perhaps some values greater or smaller depending on the inequality, to confirm the full range of solutions.

Isolation of Variables

The isolation of variables is an essential strategy in solving inequalities, as it helps simplify and rearrange the inequality to make identifying solutions easier.

This involves rearranging the inequality so that the variable you are solving for is by itself, typically on one side. To do this, use basic algebraic principles such as:

• Adding or subtracting numbers.
• Undoing any multiplication or division affecting the variable.

In the given inequality, \(22 \leq -15 + y\), by adding 15 to both sides, we remove the constant from \(-15 + y\). This simplifies things to \(37 \leq y\), which can easily be interpreted.

During variable isolation, if you ever multiply or divide by a negative number, remember to flip the inequality sign to maintain a true statement. Being mindful of these steps ensures you correctly represent the solution set.