Problem 24 - Prealgebra

Question

Solve each inequality and check your solution. Then graph the solution on a number line. $$3 x-2>10-x$$

Step-by-Step Solution

Answer

The solution is \(x > 3\).

Step 1: Simplify Both Sides

Start by simplifying both sides of the inequality. Combine like terms if possible. Here, the inequality is given as \(3x - 2 > 10 - x\). Move all terms involving \(x\) to one side of the inequality by adding \(x\) to both sides: \(3x + x - 2 > 10\).

Step 2: Combine Like Terms

Now that we have \(3x + x - 2 > 10\), combine the terms with \(x\): \(4x - 2 > 10\).

Step 3: Isolate the Variable Term

Next, isolate the \(x\) term by adding 2 to both sides: \(4x - 2 + 2 > 10 + 2\), which simplifies to \(4x > 12\).

Step 4: Solve for x

Now, divide both sides by 4 to solve for \(x\): \(\frac{4x}{4} > \frac{12}{4}\). Simplifying the fractions gives \(x > 3\).

Step 5: Check the Solution

Choose a value greater than 3, such as 4, and substitute it back into the original inequality to check: \(3(4) - 2 > 10 - 4\). This simplifies to \(12 - 2 > 6\), which is true. Thus, \(x > 3\) is indeed the correct solution.

Step 6: Graph the Solution

To graph \(x > 3\) on a number line, draw a number line, locate the point 3, and draw an open circle at 3 to indicate that 3 is not included. Shade the area to the right of 3 to represent all numbers greater than 3.

Key Concepts

Solving Inequalities

When solving inequalities, our primary goal is to find the range of values that satisfy the inequality condition. For instance, if we have the inequality \(3x - 2 > 10 - x\), we need to isolate \(x\) to solve it:

• Simplify both sides: Start by moving all \(x\) terms to one side of the inequality, which is achieved by adding \(x\) to both sides, resulting in \(3x + x - 2 > 10\).
• Combine like terms: Combine the \(x\) terms to simplify further, leading to \(4x - 2 > 10\).
• Isolate \(x\): Add 2 to both sides, giving \(4x > 12\).
• Solve for \(x\): Divide both sides by 4, resulting in \(x > 3\).

Solving an inequality involves multiple steps, much like solving an equation, but remember that inequalities also have distinctive features, like direction. Keep in mind that multiplying or dividing an inequality by a negative number reverses its sign. Fortunately, in this exercise, there was no negative coefficient to worry about. By understanding these steps, you'll be able to solve most inequalities confidently!

Graphing Inequalities

Graphing inequalities is a visual way to represent solutions. Once the inequality equation is solved, such as \(x > 3\), you can graph it on a number line:

• Draw a number line: Include a point for the relevant number, here 3, on the line.
• Open circle at 3: Since 3 itself is not a solution (the inequality is strictly greater than 3), use an open circle at 3.
• Shade the relevant area: Shade or draw an arrow starting from the open circle, extending to the right to show all numbers greater than 3 are solutions.

Graphs help to visually confirm that you've found the correct solution range. Remember, open circles denote that a particular value isn't included in the solution, whereas closed circles mean the value is included. Practicing this representation aids in understanding and checking solutions!

Checking Solutions

Checking solutions for inequalities ensures their accuracy. After arriving at \(x > 3\) as the solution, always pick a number greater than 3 to verify:

• Choose a number: Pick an easy number greater than 3, like 4.
• Substitute back: Plug 4 into the original inequality, \(3(4) - 2 > 10 - 4\).
• Simplify: Calculate both sides: 12 - 2 = 10 on the left and 6 on the right, confirming that 10 > 6, which is true.

Checking reinforces the correctness of your solution, and practicing this makes you adept at identifying any mistakes. If the inequality holds for the value you chose, then your solution is likely correct. Make this a habit to boost your confidence in problem-solving!

Prealgebra

Prealgebra skills form the foundational bedrock for algebra. They include basic arithmetic and initial exposure to algebraic concepts, like solving inequalities. Here’s why prealgebra is crucial:

• Building Blocks: Before tackling complex algebra, understanding prealgebra is essential to develop problem-solving skills.
• Introduction to Variables: Prealgebra introduces working with variables, laying the groundwork for solving equations and inequalities.
• Real-life Applications: Skills learned in prealgebra are often applicable in real life, providing practical knowledge to deal with everyday calculations.

Grasping prealgebra concepts, such as isolating variables and solving basic inequalities, prepares students for more advanced topics. Mastery at this stage leads to a smoother transition into algebra and beyond. Encouraging a strong foundation in prealgebra builds confidence and sets the path for academic success in mathematics!