How do you solve and write the following in interval notation: #4x^2 + x ≤ 3#?

Answer 1

Solution is #-1<=x<=3/4#.

#4x^2+x<=3# can be written as #4x^2+x-3<=0#

Now factorizing LHS, we get

#4x^2+4x-3x-3<=0#
or #4x(x+1)-3(x+1)<=0#
or #(4x-3)(x+1)<=0#
As equality sign is fulfilled by #x=-3/4# and #x=-1# and these are part of solution as we already have equality sign.

These two points divides number line in three regions

A- First region is #x<-1# - In this region both #(4x-3)# and #(x+1)# are negative and as such #4x^2+x-3>0#, hence this is not a solution.
B- Second region is #-1
C- Third region is #3/40#, hence this is a not a solution.
Hence solution is #-1<=x<=3/4#.
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Answer 2

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.

  1. Start by rewriting the inequality in standard quadratic form: (4x^2 + x - 3 \leq 0).
  2. Next, factor the quadratic expression: ((4x - 3)(x + 1) \leq 0).
  3. 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).
  4. Plot these critical points on a number line: (-1) and (\frac{3}{4}).
  5. 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).
  6. 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).
  7. 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).
  8. Write the solution in interval notation: ([-1, \frac{3}{4}]).
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Answer from HIX Tutor

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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