Question

solve for x and graph the solution on the number line below. -3 < x/-3 answer attempt 1 out of 4 inequality notation: number line: -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 click and drag to plot line. submit answer

Explanation:

Step1: Multiply both sides by -3

When multiplying both sides of an inequality by a negative number, the inequality sign flips. So we have \(-3\times(-3) > \frac{x}{-3}\times(-3)\). Calculating the left side: \(-3\times(-3)=9\). The right side simplifies to \(x\) (since \(\frac{x}{-3}\times(-3)=x\)). So the inequality becomes \(9 > x\) or \(x < 9\)? Wait, no, wait. Wait, original inequality is \(-3 < \frac{x}{-3}\). Let's do it again. Multiply both sides by -3. The rule is when you multiply or divide an inequality by a negative number, the direction of the inequality sign reverses. So:

\(-3\times(-3) > \frac{x}{-3}\times(-3)\)

Left side: \(-3\times(-3) = 9\)

Right side: \(\frac{x}{-3}\times(-3) = x\)

So the inequality is \(9 > x\), which is the same as \(x < 9\)? Wait, no, wait. Wait, original inequality: \(-3 < \frac{x}{-3}\). Let's check with a number. Suppose x is 0. Then \(\frac{0}{-3}=0\), and -3 < 0 is true. But if x is 9, \(\frac{9}{-3}=-3\), and -3 < -3 is false. If x is 8, \(\frac{8}{-3}\approx -2.666\), and -3 < -2.666 is true. Wait, maybe I made a mistake. Wait, let's solve it again.

Original inequality: \(-3 < \frac{x}{-3}\)

Multiply both sides by -3 (remember to reverse the inequality sign):

\(-3\times(-3) > x\)

Because multiplying by a negative number reverses the inequality. So \(-3\times(-3)=9\), so \(9 > x\), which is \(x < 9\)? Wait, no, wait. Wait, let's do it step by step.

Starting with \(-3 < \frac{x}{-3}\)

Multiply both sides by -3:

\(-3\times(-3) > \frac{x}{-3}\times(-3)\)

Left side: 9

Right side: x

So \(9 > x\), which is \(x < 9\). Wait, but let's test with x = 0. Then \(\frac{0}{-3}=0\), and -3 < 0 is true. If x = 10, \(\frac{10}{-3}\approx -3.333\), and -3 < -3.333 is false (since -3 is greater than -3.333). If x = 8, \(\frac{8}{-3}\approx -2.666\), and -3 < -2.666 is true. So that works. Wait, but maybe I messed up the sign. Wait, let's do it again.

Original inequality: \(-3 < \frac{x}{-3}\)

Let's rewrite \(\frac{x}{-3}\) as \(-\frac{x}{3}\). So the inequality is \(-3 < -\frac{x}{3}\)

Now, add \(\frac{x}{3}\) to both sides:

\(-3 + \frac{x}{3} < 0\)

Then add 3 to both sides:

\(\frac{x}{3} < 3\)

Multiply both sides by 3:

\(x < 9\)

Yes, that's correct. So the solution is \(x < 9\)? Wait, no, wait. Wait, when we multiplied by -3, we had to reverse the inequality. Let's check with x = 9: \(\frac{9}{-3} = -3\), so -3 < -3 is false. If x = 10: \(\frac{10}{-3} \approx -3.333\), -3 < -3.333 is false. If x = 8: \(\frac{8}{-3} \approx -2.666\), -3 < -2.666 is true. So yes, x < 9. Wait, but wait, let's do the first method again. Original inequality: \(-3 < \frac{x}{-3}\). Multiply both sides by -3:

Left side: -3 * -3 = 9

Right side: (x / -3) * -3 = x

But since we multiplied by a negative number, the inequality sign flips. So the original < becomes >. So:

9 > x

Which is the same as x < 9. So that's correct.

Step2: Write the inequality notation

The solution is \(x < 9\). Wait, no, wait. Wait, let's check again. Wait, maybe I made a mistake in the sign. Let's take x = 0: \(\frac{0}{-3}=0\), -3 < 0: true. x = 9: \(\frac{9}{-3}=-3\), -3 < -3: false. x = 10: \(\frac{10}{-3}\approx -3.333\), -3 < -3.333: false. x = -9: \(\frac{-9}{-3}=3\), -3 < 3: true. So x can be -9, 0, 8, etc., as long as x < 9? Wait, no, wait, when x is -9, \(\frac{-9}{-3}=3\), and -3 < 3 is true. When x is 9, \(\frac{9}{-3}=-3\), and -3 < -3 is false. When x is 10, \(\frac{10}{-3}\approx -3.333\), and -3 < -3.333 is false (because -3 is greater than -3.333). So the solution is x < 9? Wait, no, wait, maybe I messed up the multiplication. Wait, let's do it again.

Original inequality: \(-3 < \frac{x}{-3}\)

Multiply both sides by -3:

\(-3 \times (-3) > x\) (because we multiply by a negative number, reverse the inequality)

So 9 > x, which is x < 9. So the inequality notation is \(x < 9\). Wait, but let's check with x = 9: \(\frac{9}{-3} = -3\), so -3 < -3 is false. x = 8: \(\frac{8}{-3} \approx -2.666\), -3 < -2.666 is true. So yes, x < 9.

Answer:

Inequality Notation: \(x < 9\)

For the number line: We would plot an open circle at 9 (since x is less than 9, not less than or equal to) and draw an arrow to the left (towards negative infinity) to represent all numbers less than 9. But since the problem asks for the inequality notation first, the answer for the inequality notation is \(x < 9\).