Exercise 12 Page 53

First, let's split the compound inequality into separate inequalities. Compound Inequality: 4 < 6 t &+ 1 ≤ 43 First Inequality: 4 < 6 t &+1 Second Inequality: 6t &+1 ≤ 43 Notice that compound inequalities written in this way are equivalent to compound inequalities that involve the word "and." 4 < 6 t+1 and 6t+1 ≤ 43Let's solve the inequalities separately.

First inequality

Inequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, you must flip the inequality sign.

This inequality tells us that 12 is less than all values that satisfy the inequality.

Note that the point on 12 is open because it is not included in the solution set.

Second inequality

Once more, we will solve the inequality by isolating the variable.

This inequality tells us that all values less than or equal to 7 will satisfy the inequality.

Note that the point on 7 is closed because it is included in the solution set.

Compound inequality

The solution to the compound inequality is the intersection of the solution sets. First Solution Set: 12 < t & Second Solution Set: t& ≤ 7 Intersecting Solution Set: 12 < t& ≤ 7 Finally, we will graph the solution set to the compound inequality on a number line.