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Question: How do you solve \(\left| {t + 4} \right| > 10\) ?...

How do you solve ∣t+4∣>10\left| {t + 4} \right| > 10 ?

Explanation

Solution

In this question, we need to solve the given inequality with absolute value. Firstly, we will define the general definition of modulus or absolute value and then we will define it in terms of the given problem. Then we find the solution to the problem for t<\-4t < \- 4 and t⩾−4t \geqslant - 4. After that we simplify and obtain the required solution.

Complete step by step answer:
Given ∣t+4∣>10\left| {t + 4} \right| > 10
We are asked to solve the above inequality.
Note that this is an inequality with absolute value, so we can solve it directly.
Firstly, we will give the general definition of absolute value.
For any real number x, the absolute value or modulus denoted by ∣x∣\left| x \right| of x is defined as,
If x⩾0x \geqslant 0, then ∣x∣=x\left| x \right| = x
If ∣x∣<0\left| x \right| < 0, then ∣x∣=−x\left| x \right| = - x
Now we apply this definition to ∣t+4∣\left| {t + 4} \right|.
If t⩾−4,t \geqslant - 4, then ∣t+4∣=t+4\left| {t + 4} \right| = t + 4
If t<\-4,t < \- 4, then ∣t+4∣=−(t+4)=−t−4\left| {t + 4} \right| = - (t + 4) = - t - 4
So to solve ∣t+4∣>10\left| {t + 4} \right| > 10, we need to consider two cases.
Case (1) : If t⩾−4,t \geqslant - 4, we have,
⇒t+4>10\Rightarrow t + 4 > 10
Subtracting 4 from both sides of the inequality we get,
⇒t+4−4>10−4\Rightarrow t + 4 - 4 > 10 - 4
⇒t+0>6\Rightarrow t + 0 > 6
⇒t>6\Rightarrow t > 6
Case (2) : If t<\-4,t < \- 4, we have,
⇒−t−4>10\Rightarrow - t - 4 > 10
Adding 4 on both sides of the inequality we get,
⇒−t−4+4>10+4\Rightarrow - t - 4 + 4 > 10 + 4
⇒−t+0>14\Rightarrow - t + 0 > 14
⇒−t>14\Rightarrow - t > 14
Multiplying by -1 on both sides of the inequality, we get,
⇒t<\-14\Rightarrow t < \- 14
Note that when we multiply by a negative number the inequality reverses.
So finally we have, 6<t6 < t and t<\-14 t < \- 14.
Thus, by solving ∣t+4∣>10\left| {t + 4} \right| > 10, we get 6<t6 < t and t<\-14 t < \- 14.

Note: Students must know the definition of modulus or absolute value.
For any real number x, the absolute value or modulus denoted by ∣x∣\left| x \right| of x is defined as,
If x⩾0x \geqslant 0, then ∣x∣=x\left| x \right| = x
If ∣x∣<0\left| x \right| < 0, then ∣x∣=−x\left| x \right| = - x
The absolute value of x is always either positive or zero, but never negative.
Also we must know how to use this definition to a given problem and simplify it. This is more important.