Question

solve the following and graph the solutions: -3x + 7 > -1 or -3x + 7 < 1 x < \frac{8}{3} or x > 2 x < \frac{8}{3} or x > 2 x < \frac{8}{3} or x > 2 x < \frac{8}{3} or x > 2

Explanation:

Step1: Solve the first inequality \(-3x + 7 > -1\)

Subtract 7 from both sides: \(-3x + 7 - 7 > -1 - 7\)
Simplify: \(-3x > -8\)
Divide both sides by -3 (and reverse the inequality sign): \(x < \frac{8}{3}\)

Step2: Solve the second inequality \(-3x + 7 < 1\)

Subtract 7 from both sides: \(-3x + 7 - 7 < 1 - 7\)
Simplify: \(-3x < -6\)
Divide both sides by -3 (and reverse the inequality sign): \(x > 2\)

Step3: Analyze the graph of \(x < \frac{8}{3}\) or \(x > 2\)

For \(x < \frac{8}{3}\), we draw an arrow to the left of \(\frac{8}{3}\) with an open circle (since the inequality is strict, \(<\)). For \(x > 2\), we draw an arrow to the right of 2 with an open circle (since the inequality is strict, \(>\)). Looking at the options, the third graph (with open circles at 2 and \(\frac{8}{3}\), arrow left from \(\frac{8}{3}\) and arrow right from 2) matches the solution \(x < \frac{8}{3}\) or \(x > 2\).

Answer:

The correct graph is the one with open circles at \(2\) and \(\frac{8}{3}\), an arrow pointing left from \(\frac{8}{3}\) and an arrow pointing right from \(2\) (the third option among the given graphs corresponding to \(x < \frac{8}{3}\) or \(x > 2\)).