Problem 7 - Prealgebra

Question

Solve each inequality. Then graph the solution on a number line. $$h+4>4$$

Step-by-Step Solution

Answer

The solution is \(h > 0\). Graph: open circle at 0, shade right.

Step 1: Isolate the Variable

To solve the inequality \(h + 4 > 4\), we need to isolate \(h\) on one side of the inequality sign. Start by subtracting 4 from both sides: \[h + 4 - 4 > 4 - 4\] Simplifying both sides, we get: \[h > 0\]

Step 2: Graph the Inequality on a Number Line

To graph \(h > 0\) on a number line, we plot a circle at 0 and shade the number line to the right of 0, indicating all numbers greater than 0. The circle around 0 is open because 0 is not included in the solution set.

Key Concepts

Understanding Prealgebra

Prealgebra is the foundational study of mathematics that sets the stage for algebra. At its core, prealgebra introduces basic mathematical concepts that are essential for solving more complex problems later on. Concepts such as integers, fractions, decimals, factors, and the manipulation of simple equations fall under this category.

Learning prealgebra provides the tools needed to understand how numbers and operations relate to each other. It often involves understanding number lines, basic operations, and simple equations or inequalities.

This foundational knowledge makes it easier to tackle problems that involve isolating variables, like in our exercise where we solve for the variable h in the inequality $h + 4 > 4$. By mastering prealgebra, students build the confidence needed to progress to algebra and beyond.

Solving Inequalities

Solving inequalities is an extension of solving equations, but with a key difference: instead of finding an exact value, you find a range of possible values. This concept introduces the idea of greater than, less than, greater than or equal to, and less than or equal to symbols.

To solve an inequality such as \(h + 4 > 4\), the aim is to isolate the variable (in this case, h) on one side of the inequality. Follow these steps:

• Subtract 4 from both sides to remove the constant from the side of the variable: \[ h + 4 - 4 > 4 - 4 \]
• This simplifies to: \[ h > 0 \]

This means that the solution includes all values of h that are greater than 0. In solving inequalities, remember that multiplying or dividing by a negative number reverses the inequality symbol.

Graphing on Number Lines

After solving an inequality, it's important to visually represent the solution. This involves graphing the inequality on a number line. For the inequality $h > 0$, the graphing process is straightforward:

• First, draw a number line with evenly spaced marks.
• Locate and mark the number 0 on the line.
• Place an open circle on 0 because 0 is not part of the solution (indicating that it is not included in the set of numbers more than 0).
• Shade the line to the right of the circle to show all numbers that are greater than 0.

Graphing makes the abstract solution of the inequality more concrete and visually accessible. It allows us to immediately "see" the range of solutions that satisfy the inequality. This is particularly useful for checking the solutions or communicating mathematical reasoning to others.