Inequalities

Given inequality:

5(4x + 6) < 25x + 10

Goal: Find the solution set of the inequality.

Step 1: Distribute the 5 on the left-hand side.

5(4x + 6) = 5 * 4x + 5 * 6 = 20x + 30

So, the inequality becomes:

20x + 30 < 25x + 10

Step 2: Move the terms with x to one side and constant terms to the other side.

20x - 25x < 10 - 30

-5x < -20

Step 3: Divide both sides by -5 and reverse the inequality sign.

x > 4

Step 4: Conclusion.

The solution set is (4, ∞).

Given inequality:

-13 ≤ x ≤ 5

Goal: Express this inequality in the form |x - a| ≤ b, and find the values of a and b.

Step 1: Understand the form |x - a| ≤ b.

The inequality |x - a| ≤ b means that x lies within a distance of b from a. This can be rewritten as:

-b ≤ x - a ≤ b

or equivalently:

a - b ≤ x ≤ a + b

Step 2: Compare the given inequality with the desired form.

We are given -13 ≤ x ≤ 5. We need to express this as x being within a certain distance from a center value. The middle of the interval [-13, 5] is:

a = (−13 + 5) / 2 = −8 / 2 = −4

Step 3: Find b.

The distance from the center value a = -4 to the endpoints is:

b = |5 - (-4)| = |5 + 4| = 9

Step 4: Conclusion.

The values of a and b are -4 and 9, respectively.

We are given the inequality: This can be rewritten as two inequalities: Now, subtract 4 from all parts of the inequality: Next, divide through by 3: Thus, the set of all satisfying the inequality is:

The correct option is (C) :