Inequalities & Absolute Value: When the Answer Is a Range

Most of algebra asks "what value makes this true?" Inequalities ask a richer question: "what range of values makes this true?" The mechanics are almost identical to solving equations, which is good news. There's just one rule that's unique to inequalities, and one idea — absolute value as distance — that turns the trickier problems into two simple ones.

This guide covers linear and compound inequalities, the flip rule everyone forgets, and the full absolute-value toolkit, with worked and practice problems matched to real test difficulty at Northside Tutoring.

Why These Matter

Inequalities model real constraints: budgets, minimum scores, maximum capacities. The tests use them in word problems ("at least," "no more than," "fewer than") and in pure-algebra solve-the-range questions. Absolute value adds a layer the SAT and ACT love because it quietly hides two cases inside one expression.

Solving Linear Inequalities

Treat the inequality sign just like an equals sign: do the same operation to both sides until the variable is alone. To solve 3x + 4 < 19, subtract 4 (3x < 15) and divide by 3 (x < 5). Done.

The Flip Rule

Here is the only rule that's different from equations: whenever you multiply or divide both sides by a negative number, reverse the inequality sign.

Dividing by −2 flips < into >. Skipping this flip is the single most common inequality error on every test.

Compound Inequalities

A compound inequality squeezes a variable between two bounds, like −3 < 2x + 1 ≤ 7. Solve it by doing the same operation to all three parts at once: subtract 1 (−4 < 2x ≤ 6), then divide by 2 (−2 < x ≤ 3).

Absolute Value: Distance From Zero

The absolute value |x| is the distance of x from zero on the number line, so it's never negative. Because two points sit a given distance from zero, an absolute-value equation usually has two solutions:

To solve |2x − 1| = 9, split into two equations: 2x − 1 = 9 (x = 5) and 2x − 1 = −9 (x = −4).

Absolute-Value Inequalities

This is where students slip. The direction of the inequality changes the shape of the answer:

Form	Becomes	In words
|x| < a	−a < x < a	between ("and")
|x| > a	x < −a or x > a	outside ("or")

Memory aid: less thand (< gives an "and" / between answer) and greator (> gives an "or" / outside answer).

Where You'll See This — Test by Test

Inequalities and absolute value are pure algebra — nothing on a reference sheet. The SAT and ACT both test the flip rule and the two shapes of absolute-value inequalities; the SSAT Upper Level keeps to basic linear inequalities.

Watch the Lesson

Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.

Worked Example Problems

These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.

Common Mistakes to Avoid

Practice Problems — You Try

Three problems below. Work each before checking the solution.

The Northside Method — How We Teach This 1-on-1

Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.

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