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Linear inequalities

Inequalities

Japanese school year: Math I

What you learn

Learn how to solve linear inequalities to determine the valid range of values, paying close attention to negative coefficients. This skill is widely applied when dealing with range conditions and constraints. Having a solid grasp of solving linear equations beforehand will make learning this topic much smoother.

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Key points

This is the fundamental rule for adding or subtracting a number cc in an inequality. Adding or subtracting the same value on both sides leaves the inequality sign unchanged.

a<b  ⟺  a+c<b+ca < b \iff a + c < b + c

This rule shows that multiplying or dividing both sides of an inequality by a positive number cc preserves the direction of the inequality sign.

a<b,  c>0  ⟹  ac<bca < b, \; c > 0 \implies ac < bc

This important rule shows that when multiplying or dividing both sides of an inequality by a negative number cc, the direction of the inequality sign is reversed.

a<b,  c<0  ⟹  ac>bca < b, \; c < 0 \implies ac > bc

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