Question

solve the inequality and graph the solution set. write the solution set in (a) set-builder notation and (b) interv (\frac{-5x + 4}{-9} > -2)
part 1 of 3
number line graph with -5, -4, -3 (open circle), -2, -1, 0, 1, 2, 3, 4, 5
part: 1 / 3
part 2 of 3
(a) the solution set in set-builder notation is (square).
math input options: ( square|square ), ( (square) ), ( square<square ), ( square>square ), ( squareleqsquare ), ( squaregeqsquare ), ( \frac{square}{square} ), ( infty ), ( -infty )

Explanation:

Step1: Multiply both sides by -9 (reverse inequality)

When we multiply both sides of an inequality by a negative number, the direction of the inequality sign reverses. So, we have:
$$\frac{-5x + 4}{-9} \times (-9) < -2 \times (-9)$$
Simplifying both sides, we get:
$$-5x + 4 < 18$$

Step2: Subtract 4 from both sides

Subtract 4 from both sides of the inequality to isolate the term with \(x\):
$$-5x + 4 - 4 < 18 - 4$$
Simplifying both sides, we have:
$$-5x < 14$$

Step3: Divide both sides by -5 (reverse inequality again)

Divide both sides by -5. Since we are dividing by a negative number, the inequality sign reverses:
$$\frac{-5x}{-5} > \frac{14}{-5}$$
Simplifying both sides, we get:
$$x > -\frac{14}{5}$$
Or, converting \(-\frac{14}{5}\) to a decimal, \(x > -2.8\). But from the graph, we can see the critical point is -3? Wait, no, let's re - check the calculation. Wait, \(\frac{-5x + 4}{-9}>-2\)

Multiply both sides by -9: when we multiply by a negative number, the inequality sign flips. So \(-5x + 4 < (-2)\times(-9)=18\)

Then \(-5x < 18 - 4 = 14\)

Then \(x >\frac{14}{-5}=-\frac{14}{5}=-2.8\). But the graph shows an open circle at -3 and the line going to the right. Wait, maybe there was a miscalculation. Wait, let's do the calculation again.

Wait, the original inequality is \(\frac{-5x + 4}{-9}>-2\)

Multiply both sides by -9: \(-5x + 4 < (-2)\times(-9)\) (because multiplying by a negative number reverses the inequality)

\(-5x + 4 < 18\)

Subtract 4: \(-5x < 18 - 4 = 14\)

Divide by -5: \(x >\frac{14}{-5}=-\frac{14}{5}=-2.8\). But the graph has an open circle at -3. Maybe the problem has a typo or my mis - reading. Wait, maybe the original inequality is \(\frac{-5x + 4}{9}>-2\)? No, the user wrote \(\frac{-5x + 4}{-9}>-2\)

Wait, let's check with \(x=-3\): \(\frac{-5\times(-3)+4}{-9}=\frac{15 + 4}{-9}=\frac{19}{-9}\approx - 2.11\), and \(-2.11>-2\)? No, \(-2.11 < - 2\). If \(x=-2.8\) (i.e., \(x =-\frac{14}{5}\)): \(\frac{-5\times(-\frac{14}{5})+4}{-9}=\frac{14 + 4}{-9}=\frac{18}{-9}=-2\). But the inequality is \(>\ - 2\), so \(x>-\frac{14}{5}\) (since when \(x =-\frac{14}{5}\), the left - hand side is equal to -2).

But the graph shows an open circle at -3 and the solution to the right. Maybe there is a mistake in the problem or my calculation. Wait, maybe the original inequality is \(\frac{-5x - 4}{-9}>-2\)? Let's try that.

\(\frac{-5x - 4}{-9}>-2\)

Multiply by -9: \(-5x - 4 < 18\)

\(-5x < 22\)

\(x>-\frac{22}{5}=-4.4\), no.

Wait, maybe the graph is just a rough sketch. Let's proceed with the correct calculation.

The solution to the inequality \(\frac{-5x + 4}{-9}>-2\) is \(x>-\frac{14}{5}\) (or \(x > - 2.8\)).

In set - builder notation, the solution set is \(\{x|x >-\frac{14}{5}\}\) (or \(\{x|x > - 2.8\}\)). But if we consider the graph with an open circle at -3, maybe there was a miscalculation. Wait, let's re - solve the inequality:

\(\frac{-5x + 4}{-9}>-2\)

Multiply both sides by -9 (inequality sign flips):

\(-5x + 4 < 18\)

Subtract 4:

\(-5x < 14\)

Divide by -5 (inequality sign flips):

\(x>-\frac{14}{5}=-2.8\)

Yes, so the set - builder notation is \(\{x|x >-\frac{14}{5}\}\) or \(\{x|x > - 2.8\}\). If we write \(-\frac{14}{5}\) as a fraction, it's \(-2\frac{4}{5}\), but in set - builder notation, we can write it as \(\{x|x >-\frac{14}{5}\}\)

Answer:

\(\{x|x >-\frac{14}{5}\}\) (or \(\{x|x > - 2.8\}\), but the fractional form is more accurate from the calculation)