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The **Addition Property of Inequality** states that adding the same quantity to both sides of an inequality does not change the direction of the inequality. For any numbers $$a$$, $$b$$, and $$c$$:

If $$a < b$$, then $$a + c < b + c$$

If $$a > b$$, then $$a + c > b + c$$

The same rule holds for $$\leq$$ and $$\geq$$. This property is used to solve inequalities in which a number has been subtracted from the variable. To isolate the variable, the same number is added to both sides — just as with equations — and the inequality symbol stays pointed in the same direction. For example, to solve $$x - 5 < 8$$, adding $$5$$ to both sides gives $$x < 13$$, and the $$<$$ symbol remains unchanged.

Addition Property of Inequality

A logistics coordinator notes that 'Shipment A' weighs less than 'Shipment B'. If an identical 50-pound weight is added to both shipments, name the specific mathematical property that ensures 'Shipment A' will still weigh less than 'Shipment B'.

Identifying Inequality Properties in Logistics

Consider the true mathematical statement $$-4 < 2$$. If we subtract $$5$$ from both sides, we get $$-4 - 5 < 2 - 5$$, which simplifies to $$-9 < -3$$. This remains a true statement. Alternatively, if we start with $$-4 < 2$$ and add $$7$$ to both sides, we get $$-4 + 7 < 2 + 7$$, which simplifies to $$3 < 9$$, another true statement. These numerical examples verify the Addition and Subtraction Properties of Inequality, demonstrating that adding or subtracting the same quantity from both sides of a true inequality always results in another true mathematical statement with the sign pointing in the exact same direction.

Example: Demonstrating the Addition and Subtraction Properties of Inequality

To solve the linear inequality $$x - \frac{3}{8} \leq \frac{3}{4}$$, the Addition Property of Inequality is used. First, add $$\frac{3}{8}$$ to both sides of the inequality to isolate the variable. The resulting statement simplifies to $$x \leq \frac{9}{8}$$. To graph this solution set on a number line, a right bracket is placed at $$\frac{9}{8}$$ to indicate the endpoint is included, and the number line is shaded to the left. In interval notation, the solution is expressed as $$(-\infty, \frac{9}{8}]$$.

Example: Solving and Graphing $$x - \frac{3}{8} \leq \frac{3}{4}$$

To solve the linear inequality $$p - \frac{3}{4} \geq \frac{1}{6}$$, apply the Addition Property of Inequality by adding $$\frac{3}{4}$$ to both sides to isolate the variable. Finding a common denominator of $$12$$ enables adding the fractions: $$\frac{1}{6} + \frac{3}{4} = \frac{2}{12} + \frac{9}{12} = \frac{11}{12}$$. This simplifies the inequality to $$p \geq \frac{11}{12}$$. Graphing this solution involves placing a left bracket at $$\frac{11}{12}$$ on the number line and shading to the right. Expressed in interval notation, the solution is $$[\frac{11}{12}, \infty)$$.

Example: Solving and Graphing $$p - \frac{3}{4} \geq \frac{1}{6}$$

To solve the linear inequality $$r - \frac{1}{3} \leq \frac{7}{12}$$, apply the Addition Property of Inequality by adding $$\frac{1}{3}$$ to both sides. Finding a common denominator allows for combining the fractions: $$\frac{7}{12} + \frac{4}{12} = \frac{11}{12}$$. Thus, the inequality simplifies to $$r \leq \frac{11}{12}$$. Graphing this on a number line requires placing a right bracket at $$\frac{11}{12}$$ and shading to the left. The solution is written in interval notation as $$(-\infty, \frac{11}{12}]$$.

Example: Solving and Graphing $$r - \frac{1}{3} \leq \frac{7}{12}$$

To translate and solve the sentence "Twenty-seven less than $$x$$ is at least 48," apply a step-by-step translation process:

1. **Translate the left side:** The phrase "Twenty-seven less than $$x$$" indicates subtraction from $$x$$, giving the expression $$x - 27$$.
2. **Translate the inequality:** The phrase "is at least" signifies a minimum limit, which corresponds to the "greater than or equal to" symbol, $$\geq$$.

Combining these elements generates the inequality $$x - 27 \geq 48$$.

To solve the inequality, isolate $$x$$ by applying the Addition Property of Inequality—add $$27$$ to both sides:
$$x - 27 + 27 \geq 48 + 27$$
$$x \geq 75$$

To represent the solution set visually on a number line, a left bracket is placed at $$75$$ to indicate it is included, and shading continues to the right. In interval notation, this is expressed as $$[75, \infty)$$.

Example: Translating and Solving 'Twenty-seven less than $$x$$ is at least 48'

If $$a < b$$, which property guarantees that $$a + 10 < b + 10$$?

If $$a < b$$, then adding a negative number $$c$$ to both sides gives $$a + c < b + c$$.

Given $$a < b$$, put the steps in order to show $$a + 5 < b + 5$$.

Adding the same number to both sides of an inequality without changing its direction is called the ______ Property of Inequality.

If $$a < b$$, then $$a + c < b + c$$ for any number $$c$$.

Match each term to its meaning for the Addition Property of Inequality.

Case context:
Start with the true inequality $$5 < 9$$. Add $$4$$ to both sides to get $$9 < 13$$. The inequality symbol did not change direction.

Question:
Read the case below, then answer the questions that follow.

Sample answer:
The Addition Property of Inequality explains this. It states that for any numbers $$a$$, $$b$$, and $$c$$: if $$a < b$$, then $$a + c < b + c$$. Adding the same number to both sides of an inequality never changes the direction of the inequality symbol.

Key points:
- Identifies the property as the Addition Property of Inequality
- States the formal rule: if $$a < b$$, then $$a + c < b + c$$ (and similarly for $$>$$, $$\leq$$, $$\geq$$)
- Explains that adding the same quantity to both sides does not change the direction of the inequality symbol
- Connects the rule back to the example: $$5 < 9$$ becomes $$9 < 13$$ after adding $$4$$ to both sides

Rubric:
Full credit: names the Addition Property of Inequality, states the formal definition (if $$a < b$$, then $$a + c < b + c$$), and explains that the inequality symbol keeps the same direction. Partial credit: names the property but gives an incomplete or incorrect definition. No credit: names an unrelated property or gives no definition.

Read the case below, then answer the questions that follow.

If $$a < b$$, and the same number $$c$$ is added to both sides, which statement is true?

Question:
If $$a < b$$, then $$a + c < b + c$$ for any number $$c$$. Name this property and explain why the inequality symbol stays the same.

Sample answer:
This is the Addition Property of Inequality. Adding the same number to both sides of an inequality does not change its direction. Since $$a$$ and $$b$$ both increase or decrease by the same amount $$c$$, their order stays the same, so $$a + c < b + c$$.

Key points:
- Names the property as the Addition Property of Inequality
- States that adding the same number to both sides does not change the inequality symbol
- Explains that both sides change by the same amount, so their order is preserved
- Connects the explanation to the example $$a + c < b + c$$

Rubric:
Full credit: names the Addition Property of Inequality and correctly explains that adding the same value to both sides keeps the inequality symbol unchanged. Partial credit: names the property but gives a weak or missing explanation, or explains the rule without naming it. No credit: incorrect property name and no valid explanation.

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