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Why Does The Inequality Sign Change When Both Sides Are Multiplied Or Divided By A Negative Number? Does This Happen With Equations?

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Iris Phillips Profile
Iris Phillips answered
To start with the easier part of the question, no this will not happen in an equation, because whatever happens, both sides of the equation will remain equal and the equal sign will be an equal sign whichever way it is flipped.

  • Switching Signs
So why does the inequality sign change? It changes because the values, or more precisely their signs, in the inequality are changed. Because of this, the inequality sign must also be changed.

When all values are turned negative, meaning negatives are either flipped or abolished, if sides were negative, and the inequality sign stays as it is, the solution will be a different range of numbers altogether, because the inequality is changed.

  • An Example
The inequality used for this example is -3a≥-6. What happens if the inequality sign does not get flipped after multiplying the inequality by -1? The new equality would look like this: 3a≥6.

In theory, a should be a≥2. This can be tested by putting it back into the original inequality, like so: -3x2≥-6. This works, but what happens if a larger number is used? Using a 3, for example, it becomes clear that this cannot be right: -3x3≥-6.

If the inequality sign had been flipped, the new inequality would look like this: 3a≤6, so a≤2. This is still right, so the next step is to try this inequality with a number smaller than 2, so: -3x1≥-6. This clearly works.

The same process applies if the original inequality is positive and changes to negative. If signs are mixed, it is important to ensure that the inequality sign always faces the same sign. For instance, in this inequality -3a≤6, the sign faces the positive side. Multiplying by -1 results in 3a≥-6. The inequality sign still faces the positive side.
Oddman Profile
Oddman answered
Multiplying (or dividing) by a negative number reflects each expression  around the origin of the number line. This reverses the left-right ordering on the number  line.  Example  1 > -2  -1 < 2    (after multiplying by -1)
For an equation, the place on the number line changes, but the equality remains.  Example  1 = 1  -1 = -1    (after multiplying by -1. Note each expression is mirrored around the origin of the number line, but each is still at the same place the other is.)

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