Inequalities, Absolute Value, and Formulas: mastery answer key

Describe all real numbers that make $x^{2} < 9$ true.

Write a complete attempt before opening the response guide.

Verify the proposed result “$x = 4$ belongs to the truth set; $x = 7$ does not.” against the original statement.

Write a complete attempt before opening the response guide.

Complete the calculation after “Substitute each candidate into $2x - 5$.” in this problem: Find the truth set of $2x - 5 \ge 9$ among $\{-3, 0, 7, 10\}$.

Write a complete attempt before opening the response guide.

Name and justify the most efficient first move, then solve: Describe all real numbers that make $x^{2} < 9$ true.

Write a complete attempt before opening the response guide.

Compare the methods used in these two cases and identify the structural reason they differ: Find the truth set of $2x - 5 \ge 9$ among $\{-3, 0, 7, 10\}$. Describe all real numbers that make $x^{2} < 9$ true.

Write a complete attempt before opening the response guide.

Create the representation most useful for checking this result: Find the truth set of $2x - 5 \ge 9$ among $\{-3, 0, 7, 10\}$. 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 “$x = 4$ belongs to the truth set; $x = 7$ does not.” 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 “Between the boundaries, squaring produces a value below $9$; outside them it produces a value above $9$.” while solving: Describe all real numbers that make $x^{2} < 9$ true.

Write a complete attempt before opening the response guide.

In this inequalities and truth sets case, change one numerical value, solve the revised problem, and identify which parts of the original method still apply: Determine whether $x = 4$ and $x = 7$ belong to the truth set of $3x + 2 < 20$.

Write a complete attempt before opening the response guide.

Connect the opening situation “Test candidate values against a safety or budget threshold.” 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.

Explain why the method for inequalities and truth sets 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—Interpret an inequality as a condition satisfied by a region of values. 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. Find the truth set of $2x - 5 \ge 9$ among $\{-3, 0, 7, 10\}$.

Write a complete attempt before opening the response guide.

Exit check: solve and verify without referring to the displayed steps. Describe all real numbers that make $x^{2} < 9$ true.

Write a complete attempt before opening the response guide.

Classify the mathematical object and requested action in this lesson case: Solve $5 - 3x \le 17$ and write the solution in interval notation.

Write a complete attempt before opening the response guide.

State the central definition behind this outcome: Apply equation-style operations while tracking order and representing the result as a set.

Write a complete attempt before opening the response guide.

Before calculating, list every sign, endpoint, unit, grouping, or domain condition that can affect: Solve $5 - 3x \le 17$ and write the solution in interval notation.

Write a complete attempt before opening the response guide.

Explain why this opening move is valid: Subtract $5$ from both sides to obtain $-3x \le 12$.

Write a complete attempt before opening the response guide.