4. Inequalities

An inequality compares two sides that are not necessarily equal, using \(<\) (less than), \(>\) (greater than), \(\le\) (at most), or \(\ge\) (at least). Instead of a single answer, the solution is a whole range of numbers.

Great news: you solve inequalities almost exactly like equations — with one special rule.

Worked example. Solve \(2x + 1 > 5\):

• Subtract 1: \(2x > 4\).
• Divide by 2: \(x > 2\).

We graph this on a number line with an open circle at 2 (because 2 itself is not included) and an arrow pointing right toward all larger numbers.

For \(\le\) or \(\ge\), use a closed (filled) circle because the endpoint IS included.

The special rule: when you multiply or divide both sides by a negative number, flip the inequality sign. Solve \(-3x > 12\): divide by \(-3\) and flip: \(x < -4\).

⚠️ Common mistake: forgetting to flip the sign. \(-3x > 12\) gives \(x < -4\), not \(x > -4\).

💡 Tip: test your answer with one number from the range. For \(x > 2\), try \(x = 3\): \(2(3)+1 = 7 > 5\). True — so the direction is right.

Enriched GED® problem-solving lab

Learning targets

• solve and graph one-variable inequalities
• reverse the comparison after multiplying or dividing by a negative
• interpret inequality solutions in context

Visual explanation

How to read this SVG. Treat the endpoint as a boundary. A strict sign excludes it, so the circle is open; an inclusive sign includes it, so the circle is filled. The shaded region represents infinitely many values, not just the endpoint marked on the number line.

Reliable problem-solving method

1. Solve as if the comparison sign were an equals sign.
2. Reverse the sign only when multiplying or dividing both sides by a negative.
3. Choose an open or closed endpoint from the original comparison.
4. Test one shaded value in the original inequality.

Worked example 1: Inclusive boundary

Problem. Solve \(5x-2\leq18\).

Solution.

1. Add 2: \(5x\leq20\).
2. Divide by positive 5, so the sign stays: \(x\leq4\).
3. Graph a closed circle at 4 and shade left.

Answer. \(x\leq4\)

Worked example 2: Negative coefficient

Problem. Solve \(-2(3x-1)>14\).

Solution.

1. Distribute: \(-6x+2>14\).
2. Subtract 2: \(-6x>12\).
3. Divide by \(-6\) and reverse the sign: \(x<-2\).

Answer. \(x<-2\)

Worked example 3: Budget constraint

Problem. A service costs \(\$12\) plus \(\$3\) per month. With at most \(\$36\), how many months \(m\) are possible?

Solution.

1. 'At most' gives \(12+3m\leq36\).
2. Subtract 12: \(3m\leq24\).
3. Divide by 3: \(m\leq8\). Context also requires \(m\geq0\).

Answer. From 0 through 8 months

Independent self-check

Mastery habit. Say what the answer means, include units when relevant, and verify it in the original condition.