Problem 39 - Prealgebra

Question

Write a multi-step inequality that can be solved by first adding 3 to each side.

Step-by-Step Solution

Answer

The inequality is \(x - 3 < 10\), and solving gives \(x < 13\).

Step 1: Start with the Inequality

Let's consider the inequality given as an example. For simplicity, we'll use the inequality \( x - 3 < 10 \). We'll solve this inequality to find the value of \(x\).

Step 2: Add 3 to Each Side

The first step to solve the inequality \( x - 3 < 10 \) is to add 3 to both sides. This will isolate the term with the variable \(x\) on one side. So, compute \((x - 3) + 3 < 10 + 3\), which simplifies to \(x < 13\).

Step 3: Solution Check

Now that we have \(x < 13\), choose a number less than 13, like 12, and substitute it into the original inequality \( x - 3 < 10 \). We find that \(12 - 3 = 9\), which is less than 10, confirming that the solution is correct.

Key Concepts

Multi-Step Inequalities

Multi-step inequalities require you to perform more than one operation to find the solution. These operations can include addition, subtraction, multiplication, or division. The purpose of solving these inequalities is similar to solving multi-step equations, which involves isolating the variable on one side to see what values it might take.

• Begin by identifying what operations are necessary to isolate the variable.
• Order of operations matters, so always reverse the order of operations (PEMDAS/BODMAS) when solving them.
• These inequalities can often involve both positive and negative values, ensuring you maintain inequality rules for each step.

Remember, the goal is to find a range of values, not a single number. This range represents all possible solutions that satisfy the initial inequality.

Addition in Inequalities

Addition is a straightforward yet crucial operation when solving inequalities. It’s often used to simplify the inequality by eliminating a number added or subtracted from the variable. Let’s see how this looks in practice:

• In the inequality \( x - 3 < 10 \), you aim to get "\( x \)" by itself on one side.
• To counteract the \(-3\), you use addition by performing \(+3\) on both sides. This keeps the inequality balanced.

So, when you add 3 to each side, as in \((x - 3) + 3 < 10 + 3\), you simplify the inequality to \(x < 13\). Every step you take must preserve the inequality's balance, ensuring the solutions remain valid. Be careful not to flip the inequality sign, which only occurs when multiplying or dividing by a negative number.

Checking Inequality Solutions

Verifying your solutions is essential in ensuring their correctness. Confirmation through validation helps prevent errors and solidifies your understanding of the steps involved.

• After solving an inequality, choose a number from the solution range to test the original inequality.
• Using our example, if \(x < 13\), select a number like 12.
• Substitute \(12\) back into the original inequality \(x - 3 < 10\): you get \(12 - 3 = 9\), which fulfills \(9 < 10\).

This demonstrates that 12 fits the condition, proving the solution works. Verification helps ensure no steps were missed or miscalculated in isolating the variable. Performing these checks builds confidence and accuracy in solving any inequality.