Question

write a justification for each step.
4.
$hj = hi + ij$
$7x - 3 = (2x + 6) + (3x - 3)$
$7x - 3 = 5x + 3$
$7x = 5x + 6$
$2x = 6$
$x = 3$

Explanation:

Step1: Segment Addition Postulate

The Segment Addition Postulate states that if \(I\) is between \(H\) and \(J\), then \(HJ = HI+IJ\).

Step2: Substitution Property of Equality

Substitute \(HJ = 7x - 3\), \(HI=2x + 6\), and \(IJ = 3x-3\) into the equation \(HJ = HI + IJ\). So, \(7x-3=(2x + 6)+(3x - 3)\).

Step3: Simplify the right - hand side

Combine like terms on the right - hand side: \((2x+3x)+(6 - 3)=5x + 3\). So, \(7x-3 = 5x+3\).

Step4: Addition Property of Equality

Add \(3\) to both sides of the equation \(7x-3 = 5x+3\). Using the addition property of equality (\(a=b\Rightarrow a + c=b + c\)), we get \(7x-3+3=5x+3 + 3\), which simplifies to \(7x=5x + 6\).

Step5: Subtraction Property of Equality

Subtract \(5x\) from both sides of the equation \(7x=5x + 6\). Using the subtraction property of equality (\(a=b\Rightarrow a - c=b - c\)), we have \(7x-5x=5x + 6-5x\), which simplifies to \(2x=6\).

Step6: Division Property of Equality

Divide both sides of the equation \(2x=6\) by \(2\). Using the division property of equality (\(a=b,c
eq0\Rightarrow\frac{a}{c}=\frac{b}{c}\)), we get \(x = 3\).

Answer:

1. Segment Addition Postulate
2. Substitution Property of Equality
3. Simplify (Combine like terms)
4. Addition Property of Equality
5. Subtraction Property of Equality
6. Division Property of Equality