Inequalities, Absolute Value, and Formulas: mixed practice

Explain why the method for interval notation and endpoint meaning is valid here and name one nearby problem where it would not apply.

Write a complete attempt before opening the response guide.

Compare the conclusions of all three worked cases with this lesson outcome—Translate among inequalities, number-line graphs, and interval notation. Explain what remains invariant across them.

Write a complete attempt before opening the response guide.

Exit check: solve and verify without referring to the displayed steps. Write $x \le -1$ or $x > 4$ in interval notation and describe the endpoints.

Write a complete attempt before opening the response guide.

Exit check: solve and verify without referring to the displayed steps. Translate $[-2, 5)$ ∩ $(1, 8]$ into a compound inequality.

Write a complete attempt before opening the response guide.

Classify the mathematical object and requested action in this lesson case: Solve $-1 \le 2x + 3 < 9$ as one compound inequality.

Write a complete attempt before opening the response guide.

State the central definition behind this outcome: Interpret and solve intersections and unions of conditions.

Write a complete attempt before opening the response guide.

Before calculating, list every sign, endpoint, unit, grouping, or domain condition that can affect: Solve $-1 \le 2x + 3 < 9$ as one compound inequality.

Write a complete attempt before opening the response guide.

Explain why this opening move is valid: Subtract $3$ from all three parts to obtain $-4 \le 2x < 6$.

Write a complete attempt before opening the response guide.

Solve $-1 \le 2x + 3 < 9$ as one compound inequality.

Write a complete attempt before opening the response guide.

Solve $3x - 2 \le 7$ or $2x + 5 > 13$.

Write a complete attempt before opening the response guide.

Solve$$-4 < 2 - 3x \le 11$$

Write a complete attempt before opening the response guide.

Verify the proposed result “$-2 \le x < 3,$ or $[-2, 3)$.” against the original statement.

Write a complete attempt before opening the response guide.

Complete the calculation after “Solve the first inequality to obtain $x \le 3$.” in this problem: Solve $3x - 2 \le 7$ or $2x + 5 > 13$.

Write a complete attempt before opening the response guide.

Name and justify the most efficient first move, then solve: Solve $-4 < 2 - 3x \le 11$.

Write a complete attempt before opening the response guide.

Compare the methods used in these two cases and identify the structural reason they differ: Solve $3x - 2 \le 7$ or $2x + 5 > 13$. Solve $-4 < 2 - 3x \le 11$.

Write a complete attempt before opening the response guide.

Create the representation most useful for checking this result: Solve $3x - 2 \le 7$ or $2x + 5 > 13$. Use an inequality, interval notation, and a number-line description; each must show the same endpoints, inclusion, and direction.

Write a complete attempt before opening the response guide.

A learner reports “$-2 \le x < 3,$ or $[-2, 3)$.” but omits the original-condition check. Explain the risk before deciding whether the result is supported.

Write a complete attempt before opening the response guide.

Repair a solution that skips “Divide all three parts by $-3$ and reverse both order relations.” while solving: Solve $-4 < 2 - 3x \le 11$.

Write a complete attempt before opening the response guide.

In this compound inequalities case, change one numerical value, solve the revised problem, and identify which parts of the original method still apply: Solve $-1 \le 2x + 3 < 9$ as one compound inequality.

Write a complete attempt before opening the response guide.

Connect the opening situation “Find values satisfying simultaneous minimum and maximum requirements.” to the algebraic structure used in the worked case. Define quantities and units before writing any equation.

Write a complete attempt before opening the response guide.