Which value satisfies 2x − 5 ≤ 3? — x = 3.2x ≤ 8 → x ≤ 4. Of the options, x = 3 ≤ 4 satisfies it. x = 4 also works, but x = 5 and x = 6 do not.
Solve: (x/3) + 2 = 7. — x = 15.x/3 = 5 → x = 15.
Represent on a number line: x ≥ 3. — Closed circle at 3, shading to the right.≥ means "greater than OR EQUAL TO", so use a closed (filled) circle at 3. Shade right (toward larger values).
Solve: |x| = 5. — x = 5 or x = −5.|x| = 5 means x = 5 or x = −5. Both values have absolute value 5.
For what value of k does 2x + k = 5 have the solution x = 3? — k = −1.Substitute x=3: 2(3)+k=5 → 6+k=5 → k=−1.
Solve: 2x + y = 10 and x − y = 2 by elimination. — (4, 2).Add: 3x=12 → x=4. Then 4−y=2 → y=2. Solution: (4,2).
Solve by substitution: 3x + 2y = 12 and x = y − 1. — (2, 3).Substitute x=y−1: 3(y−1)+2y=12 → 5y−3=12 → y=3. Then x=3−1=2. Solution: (2,3).
To use elimination on 2x+3y=7 and 3x+5y=11, you might multiply the first equation by: — 3.Multiply eq1 by 3 and eq2 by 2 to get 6x terms that cancel. Or multiply by 3 and −2.
Does the system 2x + y = 5 and 4x + 2y = 8 have a solution? — No solution (parallel lines).Divide eq2 by 2: 2x+y=4. But eq1: 2x+y=5. Same left side, different right → contradiction. No solution.
The solution to a system represents: — The point(s) where both equations are satisfied simultaneously.The solution is the value(s) of x and y that make BOTH equations true at the same time.
Solve: x + 2y = 8 and 2x + y = 7. — (2, 3).Eliminate x: multiply eq1 by −2: −2x−4y=−16. Add to eq2: −3y=−9 → y=3. Then x=8−6=2. Solution: (2,3).
Solve: 2x + 3y = 0 and x = 3. — (3, −2).Substitute x=3: 6+3y=0 → y=−2. Solution: (3,−2).
A system is "consistent and independent" when it has: — Exactly one solution.Consistent (has solutions) and independent (not dependent/same line) means exactly one unique solution.
Use the quadratic formula on x² − 5x + 6 = 0. What is a, b, c? — a=1, b=−5, c=6.Matching to ax²+bx+c: a=1, b=−5, c=6.
Solve using the quadratic formula: x² + 4x − 5 = 0. — x = 1 and x = −5.b²−4ac = 16+20 = 36. x = (−4±6)/2. x=(−4+6)/2=1 or x=(−4−6)/2=−5.
Solve by completing the square: x² + 6x = 7. — x = 1 and x = −7.x²+6x+9=16 → (x+3)²=16 → x+3=±4 → x=1 or x=−7.
How many real solutions does x² + 4 = 0 have? — 0.x² = −4. No real number squared equals −4. Zero real solutions (two imaginary solutions).
What are the roots of 2x² − 8 = 0? — x = ±2.2x² = 8 → x² = 4 → x = ±2.
Which approach is BEST when the quadratic does not factor over integers? — Quadratic formula.The quadratic formula always works regardless of whether the quadratic factors nicely over the integers.
Solve: (x+3)² = 25. — x = 2 and x = −8.x+3 = ±5. x=5−3=2 or x=−5−3=−8.
Completing the square on x² + 8x: what number must be added? — 16.Take half the coefficient of x (8/2=4) and square it: 4²=16. Add 16 to complete the square.
Solve: x² − 2x − 15 = 0 by factoring. — x = −3 and x = 5.Need numbers multiplying to −15, adding to −2: 3 and −5. (x+3)(x−5)=0 → x=−3 or x=5.
For y = x² − 4x + 3, find the vertex. — (2, −1).x = −(−4)/(2) = 2. y = 4−8+3 = −1. Vertex: (2, −1).
A parabola opens downward. What does the vertex represent? — The maximum value of the function.When a < 0, the parabola opens downward. The vertex is the maximum (highest) point.
Sketch y = (x − 2)² − 3. What is the vertex? — (2, −3).This is vertex form y = a(x−h)²+k with h=2, k=−3. Vertex: (2, −3).
y = −x² + 4x − 1. Does the parabola open upward or downward, and what is the y-intercept? — Down, y-int = −1.a = −1 < 0 → opens downward. y-int = c = −1 (set x=0).
The x-intercepts of y = (x−1)(x−5) are: — x = 1 and x = 5.Set y=0: (x−1)(x−5)=0 → x=1 or x=5.
For y = x²−6x+9, the x-intercept(s) are: — x = 3 only.y = (x−3)². Setting y=0: (x−3)²=0 → x=3. One repeated x-intercept (vertex on x-axis).
Solve A = (1/2)bh for h. — h = 2A/b.2A = bh → h = 2A/b.
Solve F = (9/5)C + 32 for C. — C = (5/9)(F − 32).F − 32 = (9/5)C → C = (5/9)(F − 32).
Solve E = mc² for m. — m = E/c².Divide both sides by c²: m = E/c².
Solve v = u + at for t. — t = (v − u)/a.v − u = at → t = (v − u)/a.
Solve s = (1/2)at² for a. — a = 2s/t².2s = at² → a = 2s/t².
Solve V = lwh for h. — h = V/(lw).Divide both sides by lw: h = V/(lw).
Solve I = PRT for P. — P = I/(RT).Divide both sides by RT: P = I/(RT).
Why is solving literal equations useful? — To rearrange formulas for use in different contexts.Solving a formula for a different variable lets you find that variable directly, given the others. Essential in science and engineering.
Solve 3x + ky = m for x. — x = (m − ky)/3.3x = m − ky → x = (m − ky)/3.
Solve for b in a/b = c/d. — b = ad/c.Cross-multiply: ad = bc → b = ad/c.