Question

Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.

Solve.

−2/5x−9<9/10

Answer 1

why do we need to multiply it by the reciprocal in step 3?

Answer 2

To solve the inequality, we need to isolate the variable x.

Step 1: Let's first simplify the expression on the left-hand side (LHS) of the inequality.

Since -2/5x is the same as (-2/5) times x, we can rewrite the inequality as:
(-2/5) x - 9 < 9/10

Step 2: Next, we need to get rid of the -9 on the LHS. To do this, we can add 9 to both sides of the inequality.
(-2/5) x - 9 + 9 < 9/10 + 9
(-2/5) x < 9/10 + 9

Simplifying further:
(-2/5) x < 9/10 + 90/10
(-2/5) x < 99/10

Step 3: Now, we need to get rid of the coefficient -2/5 in front of x. To do this, we can multiply both sides of the inequality by the reciprocal of -2/5, which is -5/2. However, since we are multiplying by a negative number, we need to flip the inequality symbol.

(-2/5) x * (-5/2) > (99/10) * (-5/2)
x > -(99/10) * (5/2)

After simplifying:
x > -99/4

So, the solution to the inequality is x > -99/4.

Answer 3

In step 3, we multiply both sides of the inequality by the reciprocal of -2/5 in order to isolate the variable x. This is a common technique used to eliminate a coefficient in front of the variable.

When we multiply both sides of an inequality by a positive number, the direction of the inequality remains the same. For example, if we multiply both sides of x < 5 by 2, we get 2x < 10, and the direction of the inequality remains the same.

However, when we multiply both sides of an inequality by a negative number, the direction of the inequality reverses. For example, if we multiply both sides of x < 5 by -2, we get -2x > -10, and the direction of the inequality changes from less than (<) to greater than (>). This is why we need to flip the inequality symbol.

In step 3, the coefficient in front of x is -2/5, which is a negative number. So, when we multiply both sides of the inequality by its reciprocal (-5/2), we need to flip the inequality symbol to maintain the correct direction.