- Basic Math and Science Algebraic Equations and Inequalities Questions and Answers Flashcards

Solve for x: |2x − 5| > 9

Answer: x 7

|2x − 5| > 9 splits into two cases: 2x − 5 > 9 → x > 7, and 2x − 5 7.

A chemist needs a solution that is at least 30% acid. She has a 20% acid solution and a 50% acid solution. If she uses 40 mL of the 20% solution, what is the minimum volume of 50% solution she must add?

Answer: 20 mL

Let v = volume of 50% solution. Total acid ≥ 0.30(40 + v): 0.20(40) + 0.50v ≥ 0.30(40 + v) → 8 + 0.5v ≥ 12 + 0.3v → 0.2v ≥ 4 → v ≥ 20 mL.

If 3^(2x−1) = 27^(x−2), what is the value of x?

Answer: 5

Rewrite 27 as 3³: 3^(2x−1) = 3^(3(x−2)) = 3^(3x−6). Set exponents equal: 2x − 1 = 3x − 6 → −x = −5 → x = 5.

Which value of k makes the system below have infinitely many solutions? 3x − ky = 6 6x − 4y = 12

Answer: k = 2

For infinitely many solutions the equations must be identical multiples. Multiply the first equation by 2: 6x − 2ky = 12. Compare to 6x − 4y = 12: we need 2k = 4, so k = 2.

The inequality (x − 3)(x + 1) ≤ 0 is satisfied for which set of x values?

Answer: −1 ≤ x ≤ 3

The product (x − 3)(x + 1) equals zero at x = −1 and x = 3, and is negative (≤ 0) between the roots. Testing x = 0: (−3)(1) = −3 ≤ 0 ✓. So the solution is −1 ≤ x ≤ 3.

A rectangle's length is 3 more than twice its width. If the perimeter is less than 54 cm, what is the largest integer width (in cm) that satisfies this condition?

Answer: 7

Let width = w, length = 2w + 3. Perimeter = 2(w + 2w + 3) = 2(3w + 3) = 6w + 6 < 54 → 6w < 48 → w < 8. The largest integer less than 8 is 7.