Question

1. 3 - x < 9 + 5x

a. make a prediction for the sign of x by comparing the expressions on either side of the inequality sign. will it include positive or negative values? explain.
b. solve for x on the open number line.

Explanation:

Step1: Isolate x - terms on one side

Add x to both sides of the inequality $3 - x<9 + 5x$. We get $3<9 + 5x+x$, which simplifies to $3<9 + 6x$.

Step2: Isolate the term with x further

Subtract 9 from both sides: $3-9<6x$, so $- 6<6x$.

Step3: Solve for x

Divide both sides by 6. Since 6 is positive, the inequality sign remains the same. We have $\frac{-6}{6}<\frac{6x}{6}$, which gives $-1 < x$.

a. Prediction for the sign of x

The left - hand side is $3 - x$ and the right - hand side is $9+5x$. The constant on the right (9) is larger than the constant on the left (3). To make the left - hand side less than the right - hand side, when we solve the inequality, we find that $x>-1$. So x will include positive values. As $x$ can take values like 0, 1, 2 etc. which are positive and also values between - 1 and 0.

b. Representation on number line

Draw an open number line. Mark a point at - 1. Since $x>-1$, we draw an open circle at - 1 (because x is not equal to - 1) and draw an arrow to the right of - 1 to represent all the values of x that satisfy the inequality.

Answer:

a. x will include positive values. When we solve the inequality, we find $x > - 1$, which includes positive values.
b. Draw an open circle at - 1 on the number line and an arrow to the right of - 1.