Given a, b, c and d are positive numbers (such that 1 < a < b), solve each of the following for x.

Question

Don't know how get the following answers:

Question: ax+b≤x−c  ANSWER: x≤(b+c)/(1-a)

Question: ax+b>bx−1  Answer: x<(b+1)/(b-a)

Question: b−ax≤c−bx  Answer: x≤(c-b)/(b-a)

 

Thank you :)

Answer 1

Rules:

1. You may add or subtract the same value from both sided of an inequality and preserve the truth of the inequality.

2. You may multiply or divide both sides of an inequality by a positive value and preserve the truth of the inequality.

3.  When you multiply or divide both sides of an inequality by a negative value, it reverses the sense of the inequality.

 

Question 1:    ax+b≤x−c

     ax + b - x ≤ -c      [Rule 1, subtract x from both sides, stays "≤"]

     ax - x ≤ -c - b       [Rule 1, subtract b from both sides]

     x(a-1) ≤ -(c+b)      [factor out x and (-1)]

     x ≤ -(b+c)/(a-1)     [Rule 1, divide both sides by (a-1), which is positive since 1<a]

                                   [also commute (c+b) to (b+c)]

     x ≤ (b+c)/(1-a)       [-(a-1) equals (1-a)]

 

Now, the strategy boils down to this:  systematically get x on the left side by adding, subtracting, multiplying, or dividing to get the other values on the right side of the inequality.  With each step, either keep the "≤" or reverse it to "≥" based on the Rule that you used.

 

Good luck !!