Very quick inequalities queston!!

[12345678]

‘Determine the values for X which X² + 10X + 38 ≥ 22'
(X + 5)² + 13 ≥ 22
(X + 5)² ≥ 9
X + 5 ≥ +/- 3
Lets do +3 first…
X + 5 ≥ 3
Therefore, X ≥ -2
My problem is when I do – 3
X + 5 ≥ -3
X ≥ -8
I understand that the inequality sign switches when you multiply or divide by a negative number; but why does it switch in this instance?
(As the answer is X ≥ - 2 and X ≤ -8)
If anybody could explain why it switches, it would be greatly appreciated

 

[srmichael]

‘Determine the values for X which X² + 10X + 38 ≥ 22'
(X + 5)² + 13 ≥ 22
(X + 5)² ≥ 9
X + 5 ≥ +/- 3 ===> Nope!
Lets do +3 first…
X + 5 ≥ 3
Therefore, X ≥ -2
My problem is when I do – 3
X + 5 ≥ -3
X ≥ -8
I understand that the inequality sign switches when you multiply or divide by a negative number; but why does it switch in this instance?
(As the answer is X ≥ - 2 and X ≤ -8)
If anybody could explain why it switches, it would be greatly appreciated

You can't take the square root like you did when it cmes to inequalities. It does not work the same as if it would have been (x + 5)² = 9

It is easier to have subtracted 22 from both sides then you would have had:

x² + 10x + 16 ≥ 0

Then you can factor the left hand side and I think the answer will be more apparent to you.

 

[DrPhil]

‘Determine the values for X which X² + 10X + 38 ≥ 22'
(X + 5)² + 13 ≥ 22
(X + 5)² ≥ 9
X + 5 ≥ +/- 3....NO
Lets do +3 first…
X + 5 ≥ 3
Therefore, X ≥ -2
My problem is when I do – 3
X + 5 ≥ -3
X ≥ -8
I understand that the inequality sign switches when you multiply or divide by a negative number; but why does it switch in this instance?
(As the answer is X ≥ - 2 and X ≤ -8)
If anybody could explain why it switches, it would be greatly appreciated

The curve is a parabola, opening upward. Thus if x is either large negative or large positive, the curve will be >22. You have to consider what happens in the lowest part of the parabola, that is, near the vertex.

Your error is in the statement following \(\displaystyle (x+5) \ge 9\). When you take the square root, you get absolute values:

\(\displaystyle |x + 5| \ge +3\)

The two cases are whether \(\displaystyle x\) is less than or greater than \(\displaystyle -5\). Work out those two to see if you get the book answere.

 

[12345678]

IF x² + 10x + 38 = 22, then:

x^2 + 10x + 16 = 0
x = -2 or x = -8

Ohh, so I simply factorise it like it is an equation then calculate the x values from the brackets? (X+8) (X+2) so X = -8 and X=-2. Even if I solve it like this, X + 8 >(or equal) to 0 so the answer still appears as X> -8 when in reality it is the other way round. Is there a technique/ method to get the inequality the correct way without trying -7 and -9?

 

[DrPhil]

Ohh, so I simply factorise it like it is an equation then calculate the x values from the brackets? (X+8) (X+2) so X = -8 and X=-2. Even if I solve it like this, X + 8 >(or equal) to 0 so the answer still appears as X> -8 when in reality it is the other way round. Is there a technique/ method to get the inequality the correct way without trying -7 and -9?

After factoring, you still have an inequality:

\(\displaystyle (x+8)(x+2) \ge 0 \)

That inequality requires that both factors have the same sign. If \(\displaystyle x < -8\), then both are negative, while if \(\displaystyle x > -2\), then both are positive. The product of two negatives (or the product of two positives) is positive.

Consider that the curve is a parabola opening upward. The part between the two roots is negative, while the parts outside the roots are positive. The sign can only change at a root. You may also confirm that by evaluating test points in three regions: below the lower root, between the two roots, and above the higher root.

[PS - all communications concerning a problem should be posted in the thread, not in IM.]

 

[12345678]

After factoring, you still have an inequality:

\(\displaystyle (x+8)(x+2) \ge 0 \)

That inequality requires that both factors have the same sign. If \(\displaystyle x < -8\), then both are negative, while if \(\displaystyle x > -2\), then both are positive. The product of two negatives (or the product of two positives) is positive.

Consider that the curve is a parabola opening upward. The part between the two roots is negative, while the parts outside the roots are positive. The sign can only change at a root. You may also confirm that by evaluating test points in three regions: below the lower root, between the two roots, and above the higher root.

[PS - all communications concerning a problem should be posted in the thread, not in IM.]

makes more sense now, cheers