Linear Inequalities Practice Problems

Q: What changes at the Hard level?

Hard problems have parentheses, like \(-5(x - 7) \le 65\), and the coefficient in front can be negative. You distribute or divide the group out first, then isolate.

Linear Inequalities Practice Problems

The Linear Inequalities lesson covers solving, graphing, and interval notation; this practice set drills the solving — especially the flip rule — with instant checking. Each problem asks for the boundary value of x.

How the difficulty levels work

Difficulty	What it drills	Example
Easy	Positive coefficient, two-step
Medium	Coefficient may be negative — watch the flip
Hard	Parentheses, coefficient may be negative

Easy keeps the coefficient of positive, so no flip is needed — pure isolate-the-variable practice.

Medium allows a negative coefficient. Dividing by it reverses the symbol; the boundary value does not change.

Hard wraps the variable in parentheses. Distribute or divide the group out first, then finish.

Using the practice problems

Pick a difficulty and a count. For each inequality, solve it and enter the boundary value of x, then:

Check Answers grades the set.
Show Answers reveals every boundary value.
Print Worksheet builds a printable page.

On paper, write the full solution — , not just — so you practice tracking the direction too.

The rules these practice problems drill

Operation on both sides	Effect on the symbol
add or subtract any number	no change
multiply or divide by a positive	no change
multiply or divide by a negative	reverse the symbol
distribute a negative through parentheses	flip the sign of every term inside

See the Linear Inequalities lesson for graphing, interval notation, and the all-reals / no-solution cases.

Common mistakes to watch for

Forgetting to flip when dividing by a negative. gives .

Flipping when you only added or subtracted. gives — no flip.

Incomplete distribution. , not .

Entering the wrong sign. The boundary value keeps the sign it computes to; gives , boundary .

Where to go next

Two linear inequalities joined together make a compound inequality — drill those with the Compound Inequalities Practice Problems. To practice writing the answers, use the Interval Notation Practice Problems. For graphing and the special cases, see the Linear Inequalities lesson.

Frequently Asked Questions

Why does the problem ask for the boundary value instead of x > 4 or x ≤ 4?+

The answer box checks one number. The boundary value — the number x is compared to after solving — is a reliable check that you carried out the algebra correctly, including any sign flip. The direction of the inequality is visible in the problem itself.

How do I enter my answer?+

Type the boundary number as a plain value — for example 4, or -3. You can also write x = 4. Do not include a or symbol.

When does the inequality symbol flip?+

Only when you multiply or divide both sides by a negative number. Adding or subtracting any number — positive or negative — never flips it. The boundary value is the same either way, which is why it is what gets checked here.

What changes at the Hard level?+

Hard problems have parentheses, like , and the coefficient in front can be negative. You distribute or divide the group out first, then isolate.

Are the boundary values always whole numbers?+

Yes. Every inequality is built backward from an integer boundary between and , so the arithmetic stays clean and you focus on the method and the flip rule.

How does this connect to compound and absolute-value inequalities?+

Both of those reduce to solving one or two linear inequalities like these. becomes ; each part is a linear inequality solved with this method.

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