Inequalities

Everything you know about solving equations transfers directly. There is exactly one new rule, and the entire topic is built around whether you remember it.

The symbols

The phrase "at least" means greater than or equal to. "At most" means less than or equal to. "More than" and "under" are strict. Word problems use those phrasings rather than the symbols, so the translation is part of the question.

Solving is the same, except once

4x - 7 < 17

• Add 7 to both sides: 4x < 24.
• Divide by 4: x < 6.

Nothing flipped, because 4 is positive.

Now the exception.

-3x > 12

Dividing both sides by -3 requires flipping the sign:

x < -4

Why it happens is worth understanding rather than memorizing, because that is what makes it stick. 5 is greater than 3. Multiply both by -1 and you get -5 and -3, and now -5 is less than -3. Multiplying by a negative reverses the order of the number line, so it reverses the comparison.

The rule, stated exactly

Flip the sign when you multiply or divide both sides by a negative number.

Not when you:

• Add a negative number to both sides.
• Subtract a number, positive or negative.
• Multiply or divide by a positive number.
• Move a term from one side to the other.

8 - 2x <= 14

• Subtract 8 from both sides: -2x <= 6. No flip; subtracting is not multiplying.
• Divide by -2: x >= -3. Flip, because -2 is negative.

Exactly one flip in that problem, at the last step.

Avoiding the flip entirely

If the rule makes you nervous, you can sidestep it. Move the variable term to whichever side keeps its coefficient positive.

-3x > 12

Add 3x to both sides: 0 > 12 + 3x. Subtract 12: -12 > 3x. Divide by 3, which is positive, so no flip: -4 > x.

Read it back: -4 is greater than x, which is the same statement as x < -4. Same answer, no flip needed, one extra step.

It is a legitimate route and worth knowing if the flip has cost you before.

Graphing on a number line

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Zoom in

Two decisions: which circle, and which direction.

The circle: open for strict inequalities, filled for "or equal to". The circle is answering one question - is the endpoint itself a solution?

The direction: shade toward the values that satisfy it. For x > 3, shade right. For x < 3, shade left.

A useful check when the variable ends up on the right: the symbol always points toward the smaller quantity. In -4 > x, it opens toward the -4, so -4 is the larger and x is the smaller, and the shading goes left from -4.

Compound inequalities

Two conditions at once, written as one statement.

-5 < 2x + 1 <= 9

This says 2x + 1 is above -5 and at most 9. Do the same thing to all three parts.

• Subtract 1 everywhere: -6 < 2x <= 8.
• Divide by 2 everywhere: -3 < x <= 4.

So x is greater than -3 and at most 4. Graphed, that is an open circle at -3, a filled circle at 4, and shading between them.

Had a division by a negative been needed here, both symbols would flip and the whole statement would reverse.

Inequalities in word problems

The wording carries the symbol.

A crew needs at least 240 sandbags and already has 85. Each additional trip brings 25. How many trips are needed?

85 + 25t >= 240, so 25t >= 155, so t >= 6.2.

Trips are whole, and 6 trips is not enough, so the answer is 7. When a real quantity has to be a whole number, round in whatever direction actually satisfies the condition, which for a minimum means rounding up even from 6.2.

What you can skip

Across the 48 questions on this topic:

• Absolute value inequalities never appear in this topic.
• Graphing on a number line comes up twice. Know that a closed dot means the endpoint is included.

Where people lose points

Forgetting the flip when dividing by a negative coefficient.

Flipping when subtracting a negative, which requires no flip.

Flipping only one symbol in a compound inequality.

Using the wrong circle. Strict gets open.

Shading the wrong way after the variable ends up on the right side.

Rounding a word-problem answer the wrong way. Test the rounded value against the original condition.

Work one in under a minute

Solve 3 - 4x >= 19 and describe the graph.

• Subtract 3: -4x >= 16. No flip.
• Divide by -4: x <= -4. Flip.

Graph: a filled circle at -4, because of the "or equal to", shaded to the left.

Check with a value. Try x = -5: 3 minus 4 times -5 is 3 plus 20, which is 23, and 23 is at least 19. It works, so the direction is right.

Where this leads

The flip rule is the only new idea here, and it reappears anywhere a negative multiplies a comparison.