Exercise 2 Page 212

We have been given four inequalities and were asked to determine which ones are equivalent. The easiest way to do this is by simplifying all of them and comparing the results. Let's look at these one at a time.

First Inequality

To simplify the first inequality, we need to utilize the Subtraction Property of Inequalities.

3x+6≤8+2x

SubIneq

LHS-2x≤RHS-2x

x+6≤8

SubIneq

LHS-6≤RHS-6

x≤2

Second Inequality

To simplify the second inequality, we need to remember that dividing by a negative number requires us to reverse the direction of the inequality sign.

5x-5≥7x-9

AddIneq

LHS+5≥RHS+5

5x≥7x-4

SubIneq

LHS-7x≥RHS-7x

-2x≥-4

DivNegIneq

Divide by (-2) and flip inequality sign

x≤2

Third Inequality

To simplify the third inequality, we need to remember that dividing by a negative number requires us to reverse the direction of the inequality sign.

12-3x≤18

SubIneq

LHS-12≤RHS-12

-3x≤6

DivNegIneq

Divide by (-3) and flip inequality sign

x≥-2

Fourth Inequality

To simplify the fourth inequality, we can make it a bit easier if we multiply everything by -2 first. This will eliminate the fraction and all of the negatives.

-2-3/2x≥-3-x

MultNegIneq

Multiply by -2 and flip inequality sign

4+3x≤6+2x

SubIneq

LHS-4≤RHS-4

3x≤2+2x

SubIneq

LHS-2x≤RHS-2x

x≤2

Results

Now, we can compare the results. 3x+6≤8+2x& ⇒ x≤2 [0.7em] 5x-5≥7x-9& ⇒ x≤2 [0.7em] 12-3x≤18& ⇒ x≥-2 [0.2em] -2- 32x≥-3-x& ⇒ x≤2 The first, second, and fourth inequalities are equivalent.

Hints

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Practice exercises

Asses your skill

First Inequality

Second Inequality

Third Inequality

Fourth Inequality

Results