Problem 14 - Prealgebra

Question

Solve each inequality. Check your answer. $$-13>9+b$$

Step-by-Step Solution

Answer

The solution is \( b < -22 \).

Step 1: Isolate the variable

We begin by isolating the variable \( b \) on one side of the inequality. The current inequality is \(-13 > 9 + b\). We want to get \( b \) by itself, so we need to subtract 9 from each side of the inequality. This gives us: \(-13 - 9 > b\).

Step 2: Simplify the inequality

Simplify the left side of the inequality. Calculate \(-13 - 9\) to get \(-22\). So the inequality simplifies to \(-22 > b\), or equivalently \(b < -22\).

Step 3: Check the solution

To verify our solution, we can pick a value for \( b \) that is less than \(-22\). Let's choose \( b = -23 \). Plugging this into the original inequality, we have: \(-13 > 9 + (-23)\). Simplifying the right side, \(9 + (-23) = -14\). We have \(-13 > -14\), which is true, confirming our solution.

Key Concepts

Solving Inequalities

Inequalities are similar to equations but instead of an equal sign "=", they use symbols like ">", "<", ">=", or "<=" to compare two expressions. In this exercise, we examine the inequality \(-13 > 9 + b\).
Solving inequalities involves finding the range of values that satisfy the inequality. Here, we're seeking values for the variable \(b\) that make the inequality true.
Consider inequalities like signposts that guide you to a range of solutions rather than just one fixed answer. Solving inequalities means you'll often end up with a broader set of solutions than you would with a simple equation.

Checking Solutions

Once you believe you've found the solution to an inequality, it's crucial to verify it. Confirming your answer helps ensure that you performed all the steps correctly.
Take a number less than \(-22\), like \(b = -23\). Substituting \(b = -23\) back into the original inequality \(-13 > 9 + b\) gives us \(-13 > -14\), which is true.

• This process strengthens your understanding and boosts your confidence in solving inequalities.
• Having a reliable way to check your work prevents mistakes from going unnoticed.

Isolate the Variable

Isolating the variable is a fundamental step in solving inequalities. By keeping the variable b by itself on one side of the inequality, you position it more clearly within the mathematical statement.
In the problem, we began with the expression \(-13 > 9 + b\). Our goal is to isolate b. By subtracting 9 from both sides, we simplify the expression to \(-13 - 9 > b\).
This gives us a clearer view of what values b might hold, showing explicitly the relationship between numbers and the variable.

Simplifying Expressions

Simplifying expressions is about making the mathematical statement easier to work with. Once we isolated the variable in \(-13 - 9 > b\), continue by performing the calculation \(-13 - 9 = -22\).
The inequality simplifies to \(-22 > b\), indicating that b must be less than \(-22\).

• Simplifying helps in better understanding and ultimately solving any mathematical inequality or equation.
• It streamlines subsequent steps and provides clarity in interpreting the relationship the inequality represents.