Question

How do you solve \[1.2\,x+8\,<\,\,9.6\] and graph the solution graph the solution on a number line.

Answer 1

Hint: While solving linear inequalities, one may add or subtract any real number to both sides of an inequality. Also we can multiply or divide both sides by any positive real number to create equivalent inequality.

Complete step by step solution:
For given inequality \[1.2x\,\,+8\,<\,9.6\]
First, subtract \[8\] from each side of the inequality to isolate the \[x\]-term white keeping the inequality balanced.
\[1.2x\,\,+\,8\,-\,8\,\,<\,9.6\,-8\]
\[\therefore \,\,\,1.2x\,\,<\,\,1.6\]
Now, divide each side of the inequality by \[1.2\] to solve for \[x\]while keeping the inequality balanced.
\[\dfrac{1.2x}{1.2}\,\,<\,\,\dfrac{1.6}{1.2}\]
\[\therefore \,\,x\,\,<\,\,\dfrac{16}{12}\]
\[\therefore \,\,x\,\,<\,\,{4}/{3}\;\]
Or \[x\,\,<\,1{}^{1}/{}_{3}\]
To graph this on the number line, we need to put a hollow point at \[1{}^{1}/{}_{3}\] on the number line, because the inequality does not contain an “or equal to” clause. Therefore, \[1{}^{1}/{}_{3}\] is not a part of the solution. Then from that point we draw an arrow to the left because the inequality is a “less than” operator.

Note: We use these properties to obtain an equivalent inequality, one with the same solution set, where the variable is isolated. The process is similar to solving linear equations. Always remember to change the inequality sign while multiplying or dividing by negative numbers.