Exercises

1. Determine whether the given value is a solution to the inequality.
   1. \(4x \lt 8\); \(x=1\)
   2. \(-3g \geq -15\); \(g=5\)
   3. \(1 \gt -2 + z\); \(z=5\)
   4. \(-6 \lt -2y\); \(y=3\)
2. Solve each inequality. Show your algebraic steps and verify your answers.
   1. \(x-3 \gt 8\)
   2. \(h + 4 \geq 5\)
   3. \(\dfrac{t}{4} \leq 2\)
   4. \(-7x \gt 14\)
   5. \(5x-2x \lt -24\)
   6. \(\dfrac{y}{3} - \dfrac{y}{4} \leq 1 + \dfrac{1}{2}\)
3. Graph, on a number line, the solution from each part of question 2.
4. Find the smallest integer value for \(x\) that satisfies each of the following inequalities.
   1. \(2x \gt 17\)
   2. \(x-11 \geq \frac{3}{2}\)
5. Find the greatest integer value for \(x\) that satisfies each of the following inequalities.
   1. \(x+4 \leq 21.5\)
   2. \(5x \lt -9\)
6. The length of a rectangle is twice its width. Suppose the width of the rectangle is \(w\) cm.

   

   1. Write an equation to represent the perimeter of the rectangle.
   2. If the perimeter of the rectangle is \(102\) cm, find its width.
   3. If the perimeter of the rectangle is less than \(140\) cm, what are the possible values of its width?
7. Consider the sequence that begins with the terms \(7, 14, 21, 28, \ldots\) The equation \(V=7n\) represents the relationship between the term number, \(n\), and the term value, \(V\).
   1. Find the smallest value for \(n\) such that \(V\) is greater than or equal to \(100\).
   2. For how many values of \(n\) is \(V\) a 3 digit integer?
8. Frederick has a part-time job that pays \($14.65\) per hour.
   1. Write an equation to represent Frederick's total earnings, \(E\), after he has worked \(h\) hours.
   2. Frederick is saving to buy a new computer, which costs \($1573.43\) including taxes. He knows to reach his goal he must work enough hours so that his total earnings is greater than the cost of the computer.

      \(E \gt 1573.43\)

      Using the equation from part a, determine how many hours Frederick will have to work so that he has enough money to buy the computer.
   3. Frederick will not actually be able to save the full \($14.65\) from every hour that he has worked. First, he will have payroll deductions and he will also need some spending money. Frederick estimates that he will be able to save \($8.75\) for every hour he works. How many hours will Frederick have to work so that he has enough money saved to buy the computer?
9. Fill in each \(\boxed{\phantom{\square}}\) with an inequality sign so that the statement is true.
   1. If \(s \geq t\), then \(s+2 \; \boxed{\phantom{\square}} \; t + 2\)
   2. If \(a \lt b\), then \(a -6 \; \boxed{\phantom{\square}} \; b-6\).
   3. If \(g \gt h\), then \(3g \; \boxed{\phantom{\square}} \; 3h\).
   4. If \(p \leq q\), then \(-5p \; \boxed{\phantom{\square}} \; -5q\).
10. 1. Write an inequality involving the variable \(k\) and addition that has \(k \gt 3\) as the solution.
    2. Write an inequality involving the variable \(k\) and multiplication that has \(k \gt 3\) as the solution.
11. If \(a \lt b\) and \(c \lt d\), is it always true that \(a-c \lt b-d\)? If not, give an example and explain why.