Question

Solve the inequality and graph the solution set. Write the solution set in (a) set-builder notation and (b) interval notation. Expressnumbers as integers or simplified fractions.2/5(3z-5)≤5

Answer 1

Explanation

Solving the inequality

$\begin{gathered} \frac{2}{5}(3z-5)\leq5 \\ \text{ Multiply by 5 from both sides} \\ 5\cdot\frac{2}{5}(3z-5)\leq5\cdot5 \\ 2(3z-5)\leq25 \\ \text{ Apply the distributive property on the left side} \\ 2\cdot3z-2\cdot5\leq25 \\ 6z-10\leq25 \\ \text{ Add 10 from both sides} \\ 6z-10+10\leq25+10 \\ 6z\leq35 \\ \text{ Divide by 6 from both sides} \\ \frac{6z}{6}\leq\frac{35}{6} \\ z\leq\frac{35}{6} \end{gathered}$

Graphing the solution set

Step 1: We draw a number line of a suitable length.

Step 2: Since 35/6 is an improper fraction, we can convert it to a mixed number.

$\frac{35}{6}=\frac{30+5}{6}=\frac{5\cdot6+5}{6}=5\frac{5}{6}$

Step 3: Since the proper fraction is 5/6 and the whole number is 5, we divide the unit between 5 and 6 into six parts and place the number on the fifth part.

Step 4: Since the symbol of the inequality is ≤, we fill the dot and draw a line from the left to the marked point.

Answer

a) set-builder notation

$\lbrace z|z\leq\frac{35}{6}\rbrace$

b) interval notation

$(-\infty,\frac{35}{6}\rbrack$