Can you please check my answers?
Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation.
x-2/x+1 <0
my answer: (-infinity,-1) u (2, infinity)
-x+8/x-3 > or equal to 0
my answer: (3,8]
2/x-2<1
my answer: (2,4)
10-2x/4x+5 < or equal to 0
my answer: (-infinity, -5/4) or [5, infinity)
10-2x=3(x-2)+4
To check your answers, we can solve the rational inequalities step by step and determine the solution sets.
1. x-2/x+1 < 0
To find the solution to this inequality, we need to determine the critical points where the inequality changes its sign.
Step 1: Identify the critical points by setting the numerator and denominator equal to zero.
x - 2 = 0 (numerator)
x = 2
x + 1 = 0 (denominator)
x = -1
Step 2: Create a sign chart by selecting test points from three regions: x < -1, -1 < x < 2, and x > 2.
Test point -2: Substitute x = -2 into the inequality, (-2 - 2)/(-2 + 1) < 0, -4/-1 < 0, 4 > 0 (True)
Test point 0: Substitute x = 0 into the inequality, (0 - 2)/(0 + 1) < 0, -2/-1 > 0, 2 > 0 (False)
Test point 3: Substitute x = 3 into the inequality, (3 - 2)/(3 + 1) < 0, 1/4 < 0, 1/4 > 0 (False)
Step 3: Analyze the sign chart to find the intervals where the inequality is true.
From the sign chart, the solutions are in the interval (-∞, -1) U (2, ∞).
Your answer of (-∞, -1) U (2, ∞) is correct!
2. -x + 8/x - 3 ≥ 0
To solve this inequality, we'll follow similar steps as before.
Step 1: Identify the critical points.
x - 3 = 0 (denominator)
x = 3
Step 2: Create a sign chart using the test points.
Test point 0: Substitute x = 0 into the inequality, -(0) + 8/(0 - 3) ≥ 0, 0 + 8/-3 < 0, -8/3 < 0, -8/3 > 0 (False)
Test point 4: Substitute x = 4 into the inequality, -(4) + 8/(4 - 3) ≥ 0, -4 + 8/1 ≥ 0, 4 ≥ 0 (True)
Step 3: Analyze the sign chart.
From the sign chart, the solution is in the interval (3, 8].
Your answer of (3, 8] is correct!
3. 2/x - 2 < 1
Proceeding to solve the inequality:
Step 1: Simplify the inequality.
2/(x - 2) < 1
Step 2: Create a sign chart using test points.
Test point 1: Substitute x = 1 into the inequality, 2/(1 - 2) < 1, 2/(-1) < 1, -2 < 1 (True)
Test point 3: Substitute x = 3 into the inequality, 2/(3 - 2) < 1, 2/1 < 1, 2 < 1 (False)
Step 3: Analyze the sign chart.
From the sign chart, the solution is in the interval (2, ∞).
Your answer of (2, 4) is incorrect. The correct solution is (2, ∞).
4. 10 - 2x/4x + 5 ≤ 0
Let's solve the inequality:
Step 1: Simplify the inequality.
(10 - 2x)/(4x + 5) ≤ 0
Step 2: Create a sign chart using test points.
Test point -2: Substitute x = -2 into the inequality, (10 - 2(-2))/(4(-2) + 5) ≤ 0, (10 + 4)/( -8 + 5) ≤ 0, (10 + 4)/(-3) ≤ 0, 14/(-3) < 0 (True)
Test point 0: Substitute x = 0 into the inequality, (10 - 2(0))/(4(0) + 5) ≤ 0, 10/5 ≤ 0, 2 ≤ 0 (False)
Test point 5: Substitute x = 5 into the inequality, (10 - 2(5))/(4(5) + 5) ≤ 0, (10 - 10)/(20 + 5) ≤ 0, 0/25 ≤ 0, 0 ≤ 0 (True)
Step 3: Analyze the sign chart.
From the sign chart, the solutions are in the intervals (-∞, -5/4) and [5, ∞).
Your answer of (-∞, -5/4) or [5, ∞) is correct!