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Math  /  Algebra

Question2. If x=−3,y=4x=-3, y=4 and z=−5z=-5, then verify each of the following: (a) ∣x+z∣≤∣x∣+∣z∣|x+z| \leq|x|+|z| (b) ly- (c) ∣xyz∣=∣x∣⋅∣y∣⋅∣z∣|x y z|=|x| \cdot|y| \cdot|z| (d) ∣xz∣=\left|\frac{\mathrm{x}}{\mathrm{z}}\right|=

Studdy Solution

STEP 1

1. We are given specific values for x x , y y , and z z .
2. We need to verify each inequality or equation using these values.
3. We will use properties of absolute values to verify the given statements.

STEP 2

1. Verify statement (a): ∣x+z∣≤∣x∣+∣z∣ |x+z| \leq |x| + |z| .
2. Verify statement (b): ∣y−z∣≤∣y∣+∣z∣ |y - z| \leq |y| + |z| .
3. Verify statement (c): ∣xyz∣=∣x∣⋅∣y∣⋅∣z∣ |xyz| = |x| \cdot |y| \cdot |z| .
4. Verify statement (d): ∣xz∣=∣x∣∣z∣ \left|\frac{x}{z}\right| = \frac{|x|}{|z|} .

STEP 3

Calculate ∣x+z∣ |x+z| and ∣x∣+∣z∣ |x| + |z| using x=−3 x = -3 and z=−5 z = -5 .
x+z=−3+(−5)=−8 x + z = -3 + (-5) = -8 ∣x+z∣=∣−8∣=8 |x+z| = |-8| = 8
∣x∣=∣−3∣=3 |x| = |-3| = 3 ∣z∣=∣−5∣=5 |z| = |-5| = 5 ∣x∣+∣z∣=3+5=8 |x| + |z| = 3 + 5 = 8
Verify the inequality:
∣x+z∣≤∣x∣+∣z∣ |x+z| \leq |x| + |z| 8≤8 8 \leq 8
The inequality holds true.

STEP 4

Calculate ∣y−z∣ |y - z| and ∣y∣+∣z∣ |y| + |z| using y=4 y = 4 and z=−5 z = -5 .
y−z=4−(−5)=4+5=9 y - z = 4 - (-5) = 4 + 5 = 9 ∣y−z∣=∣9∣=9 |y - z| = |9| = 9
∣y∣=∣4∣=4 |y| = |4| = 4 ∣z∣=∣−5∣=5 |z| = |-5| = 5 ∣y∣+∣z∣=4+5=9 |y| + |z| = 4 + 5 = 9
Verify the inequality:
∣y−z∣≤∣y∣+∣z∣ |y - z| \leq |y| + |z| 9≤9 9 \leq 9
The inequality holds true.

STEP 5

Calculate ∣xyz∣ |xyz| and ∣x∣⋅∣y∣⋅∣z∣ |x| \cdot |y| \cdot |z| using x=−3 x = -3 , y=4 y = 4 , and z=−5 z = -5 .
xyz=(−3)×4×(−5)=60 xyz = (-3) \times 4 \times (-5) = 60 ∣xyz∣=∣60∣=60 |xyz| = |60| = 60
∣x∣=3 |x| = 3 ∣y∣=4 |y| = 4 ∣z∣=5 |z| = 5 ∣x∣⋅∣y∣⋅∣z∣=3×4×5=60 |x| \cdot |y| \cdot |z| = 3 \times 4 \times 5 = 60
Verify the equality:
∣xyz∣=∣x∣⋅∣y∣⋅∣z∣ |xyz| = |x| \cdot |y| \cdot |z| 60=60 60 = 60
The equality holds true.

STEP 6

Calculate ∣xz∣ \left|\frac{x}{z}\right| and ∣x∣∣z∣ \frac{|x|}{|z|} using x=−3 x = -3 and z=−5 z = -5 .
xz=−3−5=35 \frac{x}{z} = \frac{-3}{-5} = \frac{3}{5} ∣xz∣=∣35∣=35 \left|\frac{x}{z}\right| = \left|\frac{3}{5}\right| = \frac{3}{5}
∣x∣=3 |x| = 3 ∣z∣=5 |z| = 5 ∣x∣∣z∣=35 \frac{|x|}{|z|} = \frac{3}{5}
Verify the equality:
∣xz∣=∣x∣∣z∣ \left|\frac{x}{z}\right| = \frac{|x|}{|z|} 35=35 \frac{3}{5} = \frac{3}{5}
The equality holds true.

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