Question

x/0.125 + 0.34 < 1
options:
○ x < 3.375
○ x < 0.254
○ x < 1.125
○ x < 0.0825
○ x < 2.125
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question 5 of 5

Explanation:

Step1: Subtract 0.34 from both sides

To solve the inequality \(\frac{x}{0.125}+ 0.34<1\), we first subtract \(0.34\) from both sides. This gives us \(\frac{x}{0.125}<1 - 0.34\). Calculating the right - hand side, \(1-0.34 = 0.66\). So the inequality becomes \(\frac{x}{0.125}<0.66\).

Step2: Multiply both sides by 0.125

Next, we multiply both sides of the inequality \(\frac{x}{0.125}<0.66\) by \(0.125\) to isolate \(x\). Using the property of inequalities (when we multiply both sides of an inequality by a positive number, the direction of the inequality sign remains the same), we get \(x<0.66\times0.125\). Calculating \(0.66\times0.125=\frac{66}{100}\times\frac{1}{8}=\frac{66}{800}=\frac{33}{400} = 0.0825\)? Wait, no, wait. Wait, \(1 - 0.34=0.66\)? Wait, no, \(1-0.34 = 0.66\) is wrong. Wait, \(1-0.34 = 0.66\)? No, \(1 - 0.34=0.66\) is correct? Wait, no, wait the original inequality is \(\frac{x}{0.125}+0.34 < 1\). Let's recalculate \(1 - 0.34\). \(1-0.34 = 0.66\)? Wait, no, \(0.34+0.66 = 1\), yes. Then \(x<0.66\times0.125\). Wait, \(0.66\times0.125=\frac{66}{100}\times\frac{1}{8}=\frac{66}{800}=0.0825\)? Wait, no, wait \(0.125=\frac{1}{8}\), so \(\frac{x}{\frac{1}{8}}+0.34 < 1\), which is \(8x + 0.34<1\). Then subtract \(0.34\): \(8x<1 - 0.34=0.66\). Then \(x <\frac{0.66}{8}=0.0825\)? Wait, no, wait I made a mistake in the first step. Wait, \(\frac{x}{0.125}=8x\) (because \(\frac{1}{0.125}=8\)). So the original inequality is \(8x+0.34 < 1\). Subtract \(0.34\) from both sides: \(8x<1 - 0.34 = 0.66\). Then divide both sides by \(8\): \(x<\frac{0.66}{8}=0.0825\)? Wait, no, \(1 - 0.34 = 0.66\)? Wait, no, \(1-0.34 = 0.66\) is correct. But wait, let's check again. Wait, maybe I misread the original inequality. Wait, the original inequality is \(\frac{x}{0.125}+0.34 < 1\). \(\frac{x}{0.125}=8x\), so \(8x+0.34 < 1\). Subtract \(0.34\): \(8x<1 - 0.34 = 0.66\). Then \(x<\frac{0.66}{8}=0.0825\). Wait, but let's check the options. One of the options is \(x < 0.0825\). Wait, but wait, maybe I made a mistake in the subtraction. Wait, \(1-0.34 = 0.66\)? Wait, \(0.34 + 0.66=1\), yes. Then \(8x<0.66\), \(x <\frac{0.66}{8}=0.0825\).

Wait, but let's do the calculation again. \(1-0.34 = 0.66\). Then \(0.66\div8\): \(8\times0.08 = 0.64\), \(0.66 - 0.64 = 0.02\), \(0.02\div8=0.0025\), so \(0.08 + 0.0025=0.0825\). So \(x < 0.0825\).

Answer:

\(x < 0.0825\) (the option with \(x < 0.0825\))