How do solve #1/4

Question

How do solve #1/4<7/(7-x)# algebraically?

Sophie Adams · Accepted Answer

#-21 lt x lt 7#

We seek #x# such that:

# 1/4<7/(7-x) #

The steps to solving any nonlinear inequality are:

The solution will be those intervals in which the function has the correct signs satisfying the inequality.

We can rearrange the equation as follows:

# 1/4 lt 7/(7-x) => 1/4 - 7/(7-x) lt 0 #

# :. { 1(7-x) - 7(4) } / { 4(7-x) } lt 0 #

# :. { 7-x - 28 } / { 7-x } lt 0 #

# :. { -x - 21 } / { 7-x } lt 0 #

# :. { x + 21 } / { x-7 } lt 0 #

Which we can write as:

# f(x) lt 0 # where #f(x) = { x + 21 } / { x-7 } #

So, the critical points where a sign change can occur, are:

# x + 21 = 0 => x = -21# # x-7 = 0 \ \ => x = 7#

We must now examine the sign of #f(x)# in each interval, partitioned by these critical points i.e:

# x lt -21; -21 lt x lt 7; x gt 7 #

The easiest way to do this is to look at the sign of each factor (or component) of #f(x)# via a sign chart

{: ( ul("factor"), ul(x lt -21), ul(-21 lt x lt 7), ul(x gt 7) ), ( x+21, -, +, + ), ( x-7, -, -, + ), ( { x + 21 } / { x-7 }, +,- ,+ ) :}

So, overall, we have established that the function, #f(x) lt 0# if #-21 lt x lt 7#, so the solution of the inequality is:

# -21 lt x lt 7 #

Xavier Alonso · Answer

undefined To solve the inequality $\frac{1}{4} < \frac{7}{7 - x}$ algebraically: 1. Multiply both sides of the inequality by $4$ to clear the fraction: $$4 \times \frac{1}{4} < 4 \times \frac{7}{7 - x}$$ $$1 < \frac{28}{7 - x}$$ 2. Cross-multiply to eliminate the fraction: $$1(7 - x) < 28$$ $$7 - x < 28$$ 3. Subtract $7$ from both sides of the inequality: $$7 - x - 7 < 28 - 7$$ $$ - x < 21$$ 4. Divide both sides of the inequality by $-1$, remember to flip the inequality sign: $$x > -21$$ So, the solution to the inequality is $x > -21$.

How do solve #1/4&lt;7/(7-x)# algebraically?

How do solve #1/4<7/(7-x)# algebraically?