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| 7.1 |
Inequalities |
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| Definition |
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The relationship between two quantities that do not have the same value.
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| Symbol |
Meaning |
| \(\gt\) |
Greater than |
| \(\lt\) |
Less than |
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| Example |
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From the number line above, \(–2\) lies to the left of \(3\).
So, \(–2\) is less than \(3\).
Thus, the inequality is \(–2 < 3\).
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| Example |
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From the number line above, \(–2\) lies to the right of \(-7\).
So, \(–2\) is greater than \(-7\).
Thus, the inequality is \(–2 > -7\).
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| Describe inequality and form algebraic inequality: |
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| Example |
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From the number line above, \(x\) is less than \(8\).
Thus, \(x\lt 8\).
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| Identify relationship: |
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| Symbol |
Meaning |
| \(\geq\) |
Greater than or equal to |
| \(\leq\) |
Less than or equal to |
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| Properties of inequalities: |
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Converse property of inequality
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| Example |
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State the converse property of inequality of \(-23\gt-32\).
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Answer: \(-32\lt-23\)
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Transitive property of inequality
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| Example |
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State the transitive property of inequality of \(-15\lt-8\lt0\).
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Answer: \(-15\lt0\)
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- The inequality symbol remains unchanged when adding or subtracting a positive or negative number to or from both sides of the inequality.
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| \(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a+c&\lt b+c.\\\\ \end{aligned}\) |
| \(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a-c&\lt b-c. \\\\\end{aligned}\) |
| \(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a+(-c)&\lt b+(-c).\\\\ \end{aligned}\) |
| \(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a-(-c)&\lt b-(-c).\end{aligned}\) |
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- The inequality symbol remains unchanged when multiplying or dividing both sides of the inequality by a positive number.
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| \(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a\times c&\lt b\times c.\\\\ \end{aligned}\) |
| \(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,\dfrac{a}{c}&\lt \dfrac{b}{c}.\end{aligned}\) |
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- The direction of the inequality symbol is reversed when multiplying or dividing both sides of the inequality by a negative number.
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| \(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a\times (-c)&\gt b\times (-c).\\\\ \end{aligned}\) |
| \(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,\dfrac{a}{-c}&\gt \dfrac{b}{-c}.\end{aligned}\) |
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Additive inverse
- When both sides of the inequality are multiplied by \(-1\), the direction of the inequality symbol is reversed.
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| \(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,-a&\gt -b.\end{aligned}\) |
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Multiplicative inverse
- When performing reciprocal of both numbers on both sides of the inequality, the direction of the inequality symbol is reversed.
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| \(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,\dfrac{1}{a}&\gt \dfrac{1}{b}.\end{aligned}\) |
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