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Chapter 1: Problem 43

Fill in the blank to form a correct inequality statement. If \(x>-4,\) then \(x+4____0\)

Short Answer

Expert verified
The blank can be filled with > to form the inequality \(x + 4 > 0\).

Step by step solution

01

Understand the original inequality

We are given that \(x > -4\). This means that the value of x is greater than -4.
02

Add 4 to both sides of the inequality

To form a new inequality, add 4 to both sides of the given inequality. So, we get: \(x + 4 > -4 + 4\).
03

Simplify the right side

Simplify the right side of the inequality: \(-4 + 4 = 0\). So, we now have: \(x + 4 > 0\).
04

Write the final inequality

From the previous step, the correct inequality statement is \(x + 4 > 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Inequalities
Solving inequalities involves finding the range of values that satisfy the given conditions. Unlike equations, inequalities show the relationship where one side is not necessarily equal to the other but could be greater than or less than. For example, solving the inequality \(x > -4\) means determining all the values of \(x\) that are greater than \(-4\). To solve inequalities effectively:
  • Understand the given inequality and what it represents.
  • Manipulate the inequality while maintaining the relationship.
  • Always perform the same operation to both sides to keep it balanced.
These steps ensure that you correctly identify all values of \(x\) that satisfy the inequality.
Addition Property of Inequalities
The addition property of inequalities states that if you add the same number to both sides of an inequality, the inequality remains true. This is a vital technique used when solving inequalities. For instance, consider the inequality \(x > -4\):
Adding 4 to both sides results in:
\[x + 4 > -4 + 4\]
This simplifies to:
\[x + 4 > 0\]
In other words, adding 4 helps transform the inequality without changing its truth. Remember, whenever you add or subtract the same number to both sides of an inequality, the inequality sign stays the same. So:
  • Always perform equal operations to both sides.
  • Check the simplified result to ensure the inequality holds true.
This keeps the inequality valid and helps you step closer to finding the solution.
Simplifying Inequalities
Simplifying inequalities is a crucial step. Simplification means reducing the inequality to its simplest form while preserving its validity. For the inequality \(x > -4\), follow these steps to simplify:
  • Identify like terms on both sides of the inequality.
  • Perform the necessary operations to isolate the variable.
  • Reduce the inequality to a simpler, equivalent form.
In our example, we started with \(x > -4\) and added 4 to both sides:
\[ x + 4 > -4 + 4 \]
This simplifies to:
\[ x + 4 > 0 \]
By simplifying, we have a clearer, more straightforward inequality. This makes it easier to understand the range of values for \(x\), ensuring that the solution is intuitive and correct.

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