Inequalities & Absolute Value Practice

What an inequality actually says

An inequality compares two amounts instead of declaring them equal, so its answer is a range rather than one value. To say x is less than 4 is to describe every number below 4 at once. Solving means narrowing that range until the unknown stands alone on one side.

Four symbols carry the whole topic. < and > exclude their endpoint; ≤ and ≥ include it. On a number line that difference becomes an open circle or a filled one, and a mark in the wrong style costs a mark even when the algebra is right. The steps mirror those for an equation, with one exception: multiplying or dividing by a negative number turns the symbol round.

An answer is written as a range, such as x < 4, and read back as a sentence: any number smaller than four. Testing one value inside the range and one outside is the quickest check that the symbol points the right way, and it catches a reversed sign before the page is finished. For x < 4, try 0 as an inside value and 5 as an outside value. Substitute each into the original inequality rather than relying only on the final line. This quick check also helps with word problems, where phrases such as "at most" include the boundary but "less than" does not.

What this ladder covers

Six stops carry the ladder from reading symbols to two-sided inequalities.

• Reading the symbols: say what <, >, ≤ and ≥ mean, and whether the number at the boundary counts.
• One-step inequalities: undo a single addition or multiplication, as in x + 3 < 7.
• Two-step inequalities: clear the constant, then the coefficient, as in 2x − 1 ≥ 9.
• The negative rule: reverse the symbol whenever a negative multiplier or divisor is removed.
• Number lines: choose an open or filled circle, then shade the side that answers the question.
• Two-sided and word problems: work all three parts of 3 ≤ x + 1 ≤ 7 together, and translate phrases such as "at least".

Worked examples

Solve x + 3 < 7. Subtract 3 from both sides: x < 4. Every number below 4 works, including 3.9 and −10, and the boundary 4 does not.

Solve 5x > 20. Divide both sides by 5: x > 4. The symbol stays as it was, because 5 is positive and dividing by it does not change the order of the two sides.

Solve −2x > 8. Divide both sides by −2 and flip the symbol: x < −4. Check with −5: −2 × −5 = 10, and 10 is greater than 8, so −5 belongs in the answer.

Solve 3 ≤ x + 1 ≤ 7. Subtract 1 from all three parts: 2 ≤ x ≤ 6. The middle part is worked on, not the outer two, so the range keeps its two limits.

Solve 2x − 1 ≥ 9. Add 1 to both sides to get 2x ≥ 10, then divide by 2: x ≥ 5. Two moves, and the symbol is untouched because both divisors were positive.

Mistakes worth watching for

One rule about negatives accounts for most of the slips on this page.

• −2x > 8 answered as x > −4. The symbol must turn when both sides are divided by a negative. The test value −5 fits the original and fails x > −4, which shows the flip is needed.
• A filled circle at 4 for x < 4. A strict inequality leaves the endpoint out, so the circle stays open. Filling it says 4 is an answer when it is not.
• Subtracting from one side only. x + 3 < 7 − 3 is not a solution, because the same operation has to reach both sides. Changing one side alters the comparison.
• Answering x + 3 < 7 with x > 4. The arithmetic gave x < 4 and the symbol was reversed for no reason. Test 0: 3 is less than 7, so the smaller numbers are the answer.
• Shading the wrong side of the boundary. For x < 4 the arrow runs left, towards the smaller numbers. Shading right describes x > 4, which is the opposite answer.
• Including an endpoint that the words exclude. "At most 4" includes 4 and uses ≤, while "less than 4" leaves 4 out and uses <. The phrase in the question decides whether the endpoint is filled.

How the ladder works

This ladder has six stops, the first a Warm-Up of five questions and the last a set of eight or nine. Clear a stop with 80% of first attempts right; nothing is locked, and each retry builds a fresh set.

Keep going

Print a set of inequalities with the inequalities worksheet generator, see the shaded region on screen with the graphing calculator, or move to linear equations, where the same balance method solves for an exact value.

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