How do you solve and write the following in interval notation: #4x^2 + x ≤ 3#?
Solution is
Now factorizing LHS, we get
These two points divides number line in three regions
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To solve the inequality (4x^2 + x \leq 3) and write the solution in interval notation, you first need to find the values of (x) that satisfy the inequality. Then, you represent the solution using interval notation.
- Start by rewriting the inequality in standard quadratic form: (4x^2 + x - 3 \leq 0).
- Next, factor the quadratic expression: ((4x - 3)(x + 1) \leq 0).
- Determine the critical points by setting each factor equal to zero and solving for (x): (4x - 3 = 0) and (x + 1 = 0).
- Solving (4x - 3 = 0), you get (x = \frac{3}{4}).
- Solving (x + 1 = 0), you get (x = -1).
- Plot these critical points on a number line: (-1) and (\frac{3}{4}).
- Test intervals between these critical points to determine where the inequality is true. Choose test points such as (x = 0), (x = -2), and (x = 1).
- Evaluate the inequality (4x^2 + x - 3 \leq 0) for each test point.
- For (x = 0), (4(0)^2 + 0 - 3 = -3), which is less than or equal to (0).
- For (x = -2), (4(-2)^2 + (-2) - 3 = 17), which is greater than (0).
- For (x = 1), (4(1)^2 + 1 - 3 = 2), which is greater than (0).
- Based on the test results, determine the intervals where the inequality is true.
- The interval ([-1, \frac{3}{4}]) satisfies the inequality (4x^2 + x - 3 \leq 0).
- Write the solution in interval notation: ([-1, \frac{3}{4}]).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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