Introducing Linear Inequalities

A one-step inequality requires a single inverse operation. A two-step inequality requires two. In both cases, the steps are identical to solving an equation, just carry the inequality sign through.

For example, $x + 8 > 15$: subtract 8 from both sides to get $x > 7$. Every number greater than 7 is a valid solution.

Inequalities are solved like equations with one extra rule: dividing or multiplying both sides by a negative number flips the inequality sign. One-step: one inverse operation. Two-step: undo addition/subtraction first, then division/multiplication.

Pause, copy the flip rule (dividing or multiplying both sides by a negative number reverses the inequality sign), the two-step solve order for inequalities (undo addition/subtraction first, then multiplication/division), and note that the flip only applies when the operation involves a negative number into your book.

We just saw one-step and two-step inequalities solved like equations, with the key extra rule that dividing or multiplying both sides by a negative number flips the inequality sign. That raises a question: what happens when the variable is inside brackets with a positive coefficient outside, does the bracket trigger the flip? This card answers it → a positive coefficient outside brackets never triggers a flip; the flip only occurs when you divide or multiply by a negative number.

Bracket inequalities such as $4(x - 1) > 12$ are solved by expanding or dividing first, then applying inverse operations. The flip rule only activates when you multiply or divide by a negative number.

A common error is forgetting to flip when rearranging a negative coefficient inequality such as $-2x \geq 10$. Dividing both sides by $-2$ flips $\geq$ to $\leq$, giving $x \leq -5$.

Bracket inequalities: expand or divide out the coefficient first, then solve. Flip rule: only flips when dividing (or multiplying) by a negative, the bracket’s positive coefficient does not trigger a flip.

Pause, copy the bracket inequality strategies: (1) divide out the positive coefficient first; (2) expand the bracket then solve, and the flip-rule condition (flip only if the bracket's coefficient is negative or you later divide by a negative) into your book.

We just saw bracket inequalities and the precise condition for the flip, only when dividing or multiplying by a negative number, not simply because brackets are involved. That raises a question: once you have a solution like x > 2, how do you verify it's correct and communicate it in a way that earns full marks? This card answers it → verify by substituting a test value from the solution set into the original inequality; communicate by writing the solution set using inequality notation and checking boundary inclusion (strict or non-strict).

After solving, always substitute a test value to verify the solution. Choose a convenient value that should be in the solution set, if the original inequality is satisfied, your answer is likely correct.

In context, state the answer as a sentence: "She needs to work at least 8 hours" rather than just writing $h \geq 8$.

After solving an inequality, verify by substituting a test value from the solution set into the original inequality, if the inequality holds, the solution is correct. State the solution set using inequality notation and check boundary inclusion.

Pause, copy the test-value verification method (substitute a value from the solution set into the original inequality; if the inequality holds, the solution is correct) and the boundary-inclusion check (strict > or <: boundary not included; non-strict ≥ or ≤: boundary included) into your book.

1. Solve $x - 3 \geq 8$.
2. Solve $5x < 35$.
3. Solve $-4x > 20$. Remember the flip rule.
4. Solve $2(x + 3) \leq 14$.